6.Mass Balancing

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Background

underflow streams. Therefore, percent solids determined from wet and dry sampleweights are preferred.

Steady State JKSimMet is a steady state simulator. Hence, models can most usefully be fitted todata which were taken at steady state. The most common approach is to take a seriesof regular samples and combine them to make composite samples which cover aperiod which is long (several hours) compared with circuit fluctuations, which mustbe kept to a minimum.

If circuit variations are a serious problem, sample and fit one process unit at a time. JKSimMet can be used to combine the units and predict circuit steady state behaviour.

Sampling Good sampling practices are a topic in themselves. Some useful references are thoseof Gy (1982) and Lyman (1986) .For a simple estimating technique for sampling requirements refer to the paper byLyman (1986).There are well established rules for calculating the accuracy of a sampling and assayprocess. These can be used to establish an error model which can then be used toprovide estimates of standard deviation for each point. Alternatively, 5 to 10 replicatesamples can be taken and processed. If these input accuracies are established, then theestimates of accuracy used for flow rates and assays will be real estimates and notrelative estimates.If replicate sampling is carried out for assays on a number of streams (i.e. a range ofassay values), a simple two term error model can be generated by plotting relativestandard deviation against average assay values from each stream.The intercept and slope of this plot will provide fixed (minimum) and relative (%)error components which can be used in the generalised version of the Whiten model.A sensible maximum (absolute) error will also need to be specified.Determining good estimates of the errors associated with each sampling point willprovide a more reliable mass balance.

Mass Balancing is a type of model fitting. The models in this case are quitefundamental. Hence, they do not impose the experience knowledge (which is builtinto other mathematical process models) onto the data.The mass balancing models are:

a stream combiner (for example, a pump sump),a general stream splitter (for example, a hydrocyclone or a flotation cell),

a unit that conserves some properties but not others(for example, a grinding mill will preserve total assays and flow rates but not sizefractions).

The basis of the mass balancing algorithms is the differences in composition of variousstreams; that is, the differences generated by the process equipment.Consider a process with these streams having assays a, b, c:

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If the solids flow rate in a stream of assay "a" is 100 tph, then:

(1)

where x is the solids flow rate in stream of assay "b" and then:

(2)

This is the basis of the traditional two-product solution, where a, b and c may beassays for a size fraction, element or any other conserved property.It does not matter what kind of assays a, b and c are, as long as there is somedifference in their values. For example, if the process is a splitter and the assays are allthe same: a = b = cand therefore, using equation (2), then x= 0/0 which is undefined.

Expressed another way, the flow rates can be estimated only if a process imposes adifference on its products; that is, some information is imparted by the process. If noinformation is imposed, as is the case with a splitter, then the information cannot beused to make estimates, as it is not there to begin with. Note that the split ratio of asplitter can be used in the mass balance.It follows that the most useful properties to use for mass balancing around a processunit will be those which have the largest difference in the product streams.This means that size assays will work well around a size classifier such as a screen or ahydrocyclone, and elemental assays will work well around a flotation circuit. Thereverse will generally not be true, with some notable exceptions. For example,elemental assays such as gold or lead are often very useful around a hydrocycloneclassifier because its density-separating characteristic will usually produce a largedifference in these assays.The power of this program lies in its ability to use a wide range of assays across a largeflowsheet. The program algorithm is driven by the assays with large differences but

still takes account of those with small differences.Concept : M assBalancing

The mass balancing module takes all selected streams and calculates the smallest set ofdata adjustments which will make the data consistent.If some (or all) of these streams are measured (sampled and sized, etc), theexperimental measurements can be compared with the data. The root mean square ofthe normalised differences between measured data and adjusted data is taken as ameasure of goodness of fit of the model.

i.e.

Hence, the mass balancing program adjusts user selected flow rates to find a best setof flow rates which make the balance output match the experimental measurements asclosely as possible.

Concept:W ei ghted Sumof Squares

If the precision of each data point is measured (or can be estimated from experience),then each difference between experimental data and mass balanced prediction isnormalised by dividing by its precision. That is, a small difference (or adjustment)between an accurate data point and its mass balanced prediction will make the samecontribution to the weighted sum of squares as a large difference from an inaccuratedata point.




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How the Mass Balancing Program Works

Concept:StandardDeviat ion

The usual measure of precision is the standard deviation. If repeated measurementsare made of any data point, experimental variations will cause variations in themeasured value x i.Then with many repeats, the mean of the values will provide an estimate of the truevalue of x.Subject to a number of assumptions, the expected variations from true x can becharacterized by one number - the standard deviation defined as:

StandardDeviation

If the measurements are normally distributed then, out of 100 measurements, 67 couldbe expected to lie within plus or minus one standard deviation of the true value (asestimated by the mean), 95 within plus or minus two standard deviations and 97within plus or minus three standard deviations.

Concept:Estimating

StandardDeviat ion

Experimentally, 5 to 10 complete observations, that is, independent sampling plusanalysis, will provide an estimate of the standard deviation. The mean of such a set ofmeasurements should provide a good test of sampling precision - if the test circuit wasat steady state.

Concept : RM S(Root M eanSquar e) Err ors

As its name suggests, the RMS (root mean square) error is the square root of theresidual mean square error, which is the error associated with the calculated massbalanced data and the experimental data.A root mean square error of value 1 means that all data points were estimated witherrors similar to the measured standard deviations of the experimental data points.Obviously, the more the RMS error varies from one, the more error is associated withfitting the data.

The mass balancing program used by JKSimMet is a program called JK2DMBalwritten and developed by Dr Stephen Gay. Other JKMRC staff, including MichaelAndrusiewicz, Jake Stoll, Ricardo Pascual and Robert Lasker, provided assistance withthe integration of JK2DMBal into the JKSimMet program.The mass balancing problem is essentially a minimisation of sum of squares withmulti-linear constraints. This corresponds to a common set of mathematical problemsgenerally described as the quadratic problem.The algorithm is based on a Quasi-Newton approach which means that the errors inthe constraints are used to determine the changes required in calculated variables,with the amount of movement of the calculated variables controlled by the standard

deviations of the experimental values.Hence standard deviation values of 0 mean that a calculated value will equal theexperimental value and will not change. If all the standard deviations are near 0, theprogram will not have enough freedom to find a solution, and therefore will notconverge. The JKSimMet standard deviation interface provides formulae to allowappropriate standard deviation values to be set.When a plant is surveyed, size and assay information are often obtained, as well aspercent solids. However, the solids flow is only known for a small number of streams(and in some cases, the feed stream only). Very high standard deviation values imply




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Learning Mass Balancing

that the corresponding experimental value is unreliable. Hence, large standarddeviation values are often used for the solids flow values.The main algorithm (Quasi-Newton algorithm) needs reasonable starting values ofsolid and water flow in order to converge satisfactorily. In order to estimate theseflows (principally where experimental values are not given), a second algorithm isused.This second algorithm uses information such as assays and size distribution data toestimate solids flows. This algorithm is very similar to the method described as theMorrison solution in JKSimMet. The main difference is that it also uses the standarddeviations of the assay and size information, and it provides an estimate of thestandard deviation of the estimated solids flow.If there are many missing streams, or missing data, the algorithm will still have somedifficulty in obtaining a solution as the standard deviation values are too high. A thirdalgorithm is used to reduce the standard deviation values as the algorithm proceeds toensure that convergence is obtained.The three algorithms (Quasi-Newton, missing flow estimation and variable standarddeviation reduction) are all integrated together within the one algorithm interfaced to

JKSimMet. Even though the algorithms work together it still remains that in somecircumstances data reduction may be required to improve performance. For example,if there are many missing solids flow values it is best to mass balance solids flow firstwith information such as sizes and head assays prior to mass balancing assays withinsize classes.

JKSimFloat has a hierarchy of data associated with the mass balancing algorithm. Thishierarchy has the solids mass flow at the top, with head assays, size fractions and %solids all subordinate to the solids mass flow.The assays of size fractions are subordinate to size fractions, and the mass flow ofwater is always a bottom level measurement.This can be represented schematically by the following diagram.

The mass balancing module of JKSimMet is useful in two areas. Firstly, it provides acheck on data accuracy that is not model dependent. The mass balancing models arecorrect (that is, they contain no built in experience). Hence, if the data balance wellbut the model fitting does not fit well, it indicates that the model is not appropriate.




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Model Types for Mass Balancing

Where coarsely sized samples are used, as in crushing and screening circuits, the massbalanced data may be more useful as the starting point for model fitting than the rawdata.Secondly, mass balancing is useful for determining flow rates and recoveries aroundcomplex circuits. The example we will use in this section, Learning Mass Balancing, isconcerned with flow rates in the comminution circuit of a copper concentrator.

NormalSequence forMass Balancingin aComminutionCircuit

It should be noted that JKSimMet V6 provides the ability to balance assays within sizeclasses as well as total assays and sizes. Unlike V5, the user does not balanceeverything at once. Under the assumption that in a comminution circuit your sizingdata will be the most reliable survey data, the suggested process to balance a wholecircuit is as follows:

1. Make selections for all required equipment, streams, sizes and elements.2. Set TPH Solids and Sizes to Adjust and all other components to Unused,

then run the balance.3. Then set the TPH Solids and Sizes to Fixed, the TPH Water to Adjust and %

Solids to Influence and run the balance again.4. Set the TPH Water and % Solids back to unused and the Elements to Adjust

and run the balance again.5. Set the Elements to Fixed and Size x Element to Adjust and run the balance a

final time.The individual steps will be covered in more detail over the remainder of this chapter.

Mass BalancingSequence whereAssay Data isthe MostReliable Data

In cases where your assay data is considered more reliable that the sizing data, youwould modify the above sequence as follows:

1. Make selections for all required equipment, streams, sizes and elements.2. Set TPH Solids and Elements to Adjust and all other components to Unused,

then run the balance.3. Then set the TPH Solids and Elements to Fixed, the TPH Water to Adjust and

% Solids to Influence and run the balance again.

4. Set the TPH Water and % Solids back to Unused and the Sizes to Adjust andrun the balance again.5. Set the Sizes to Fixed and Size x Element to Adjust and run the balance a

final time.

Mass BalancingSequence where% Solids Data isthe MostReliable Data

In cases where your % solids data is considered the most reliable data, you would bebetter using the balance sequence below:

1. Make selections for all required equipment, streams, sizes and elements.2. Set TPH Solids and TPH Water to Adjust and % Solids to Influence and all

other components to Unused, then run the balance.3. Now set the TPH Solids to Fixed, TPH Water and % Solids to Unused and

the Sizes to Adjust and run the balance again.

4. Set the Sizes to Fixed and the Elements to Adjust and run the balance again.5. Set the Elements to Fixed and Size x Element to Adjust and run the balance a

final time.

In V6 of JKSimMet the flowsheet drawing for mass balancing is the same one used forsimulation and model-fitting, with the full range of equipment icons available to draw




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Entering the Data

the circuit diagram. However, no matter what equipment icon is visible, there areonly two model types in mass balancing. These are:

classifier or mixer unitThis unit either selects particles to go to different product ports of the unit(classifier) or adds particles from different feeds (mixer). That is, particlesare sorted or mixed in this type of unit, not broken down or altered.

transform unitIn this unit assays are preserved but size structures are destroyed. In massbalancing all comminution devices are transform units.

The mass balance algorithm decides which type of mass balance unit is requiredaccording to the flowsheet icon selected by the user.

Data EntryAlternatives

The first step in a mass balancing exercise is to enter the experimental data. This canbe performed in two ways:

Individual St ream Dat a windows, orConfi gurable Str eam O verv i ew

Both methods have their advantages and disadvantages; it is mainly dependent onwhat form the experimental data is in.

Accessing theIndividualStream Data

The Streams window provides access to the Stream data windows to view the data foreach stream. The Streams window for each stream can be opened by three methods:

1. Double-clicking on a stream in the flowsheet.2. Selecting Stream on the Flowsheet sub menu or3. Clicking on the Stream icon.

All of the streams in the current flowsheet may be accessed from the Streams window

via the Stream Name drop-down list. The Streams window can only be accessed whenthe flowsheet is locked. A typical Streams window is shown below.The Streams window also shows the equipment from which the stream originates andthe equipment to which the stream flows. Clicking on the double arrow buttons nextto the From Equipment and To Equipment dialogue boxes opens the Equipmentwindow relating to that equipment.

The St ream dat a window is accessed by clicking on the double arrow button next tothe selected stream. Once the Str eam dat a window has opened, the user can view thevarious categories of data by selecting the appropriate tab. The name of the streambeing viewed is displayed in the header of the Str eam dat a window. The Streamswindow remains open and accessible after the required Stream data window hasopened. If you then select another stream from the drop-down and click again on the




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double arrow, you will open a second Stream data window for the new stream. Youcan open as many Stream data windows as you require in this way.

A typical Stream data window is shown below.

The status of the stream can be set to Major, Minor, or Missing from the drop-down box at top right. See later in this help file for an explanation of these terms.

Totals Tab The Totals tab lists the overall properties of the stream. The other tabs and the datadisplayed are:

Name of Tab Data Displayed

Sizing data Size fractions and the proportion of the stream ineach size fraction

Elemental assay data Elements and the proportion of each element in thestream

Size by elemental assaydata

Proportion of each size class that is made up of eachelement

Data can be inserted into the individual stream windows using the copy and pastefunctions from Excel.

Sizing Data If you click on the Sizing Data tab you will here be able to enter the sizing data for thesize fractions that have been defined for this project and the associated SD values. TheStream Data window will then look similar to the one shown below:




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The above window will also show balanced values from the most recent balance,provided the balancing engine has been run at this point.

Elemental AssayData

The next tab is for the Elemental Assay Data and the appearance of this window willbe similar to the one shown below when this tab has been selected.

The last tab is for Size by Elemental Assay Data and a screen grab of the appearancewith this tab selected is shown below:




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Standard Deviation Calculations

In this case the numbers are all zeros since no sizing data have been entered at thispoint.

ConfigurableStreamOverview

An alternative to entering data individually into the stream windows is to use theConfigurable Stream Overview. This is recommended when the data has beeninitially configured into a similar style table in Excel. Such a table can then simply becopied and pasted into JKSimMet.Click on the Configurabl e Str eam O vervi ew icon or select this item from the Flow sheet menu to open the Confi gurable Str eam O verv i ew window. You may need to create anew overview that will have the data in the same order as it is structured within theExcel source file. Alternatively you could manipulate the setting out of the Excel file tomatch a structure you have previously created within the Confi gurable St reamOverview . For more information on how to make changes or create new StreamOverviews, you should revise the topic Using the Configurable Stream Overview .

Every data point must have a standard deviation entered, even if it is Missing.During the mass balancing routine, if any data point has 0 as its standarddeviation, an error message will be displayed with an option for the user toenter the correct SD.Standard deviations can be entered individually in the stream data windows, orthrough the Confi gurable St ream O vervi ew window. They can also be copiedand pasted from Excel.

Entering a standard deviation in individual stream data windows isparticularly useful if, for instance, no experimental data for a particular streamexists. The procedure for this is outlined below.

Entering IndividualStream StandardDeviations

Ensure the flowsheet is locked and double click on a stream to bring up thestream window for that stream.If the sample or flow data for a particular stream is not available, you can leavethe experimental value set to zero and then set the standard deviation value toMissing.




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If missing is entered for the standard deviation, the mass balancer willrecognise that no experimental data was recorded for this stream. However, thestream will be included in the mass balance and a balanced value will begenerated using the rest of the data set.For an initial pass, the actual stream can be omitted from the balance routine,rather than setting the SD values to missing. This will not make muchdifference in a simple flowsheet, but may speed up the balance in a morecomplex one.

Alternatively, if experimental data exists for that stream and the standarddeviation is known, it is possible to enter that standard deviation directly inthis window. Simply click in the appropriate cell of the SD column and type inthe standard deviation. Press Enter when complete.It is also possible to copy and paste standard deviation values from Excel intothe individual stream data windows and the Configurabl e Str eam O vervi ew window.

The Aut omati c SD Calculat ion window allows the user to set SD values formultiple streams and data points simultaneously. This is accessed by the SDCalculation button in both the individual stream window and the ConfigurableSt ream O verv i ew windows.

Entering StandardDeviations for allStreams

The Aut omati c SD Calculat ion window appears as follows.




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The left-hand panel lists all the streams in the current flowsheet. Individual

streams can be selected by clicking on the stream name. Multiple streams canbe selected in a similar manner to the way this is achieved when usingWindows Explorer. A contiguous group of streams can be selected by holding when clicking on a stream further down the list. A selection of noncontiguous streams can be made by holding down and clicking on thestreams required.The Property drop-down permits the user to select the parameter that is tohave the standard deviation formula applied to it. To select a property, click thedrop-down arrow and double click on the desired property.The SD Calculation Option drop-down allows the user to choose from a listof mathematical formulae and other options for setting the standard deviations.This list contains the following formula options: Bounded percentage,

Parabolic, SD Multiplication Factor, Exp Val Multiplication Factor,Replacement Value and Missing Value.If you tick the check-box labelled Include SD values currently set to missing?the program will take the experimental value for the stream and apply theselected SD calculation to this value.There are also default values set to enable rapid SD calculation:

Fixed : SD = 0.001Poor : SD = 20% of the experimental valueAverage : SD = 10% of the experimental valueGood : SD = 5% of the experimental valueExcellent : SD = 2% of the experimental value

These values can also be changed if necessary in the Aut omati c SD Calculat ion window.Note that although this facility allows you to use relative SDs as a quick way ofsetting up the balance, their use is not generally recommended because abalance using relative SDs takes most notice of the smallest assay values andthese are often the least well defined.

Bounded Percentage When the Bounded Percentage option is selected, the user inputs the Upper(U), Lower(L) SD limits and the percentage error in between (P) to calculate thestandard deviations for an experimental data value (x).




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Preparation for Mass Balancing

Parabolic When the Parabolic option is selected, the user inputs the proportionalityconstant (a) which will be used to calculate the magnitude of the standarddeviations associated with each experimental data value (x).

SD MultiplicationFactor

When the SD Multiplication Factor option is selected, the user inputs themultiplication factor to be multiplied by the existing standard deviations in thesystem.

Exp ValueMultiplication Factor

When the Exp Value Multiplication Factor option is selected, the user inputsthe multiplication factor to be multiplied by the existing experimental values inthe system.

Replacement Value When the Replacement Value option is selected, the user is asked to input thevalue to replace selected SD values in the system.

Missing Value When the Missing Value option is selected the user does not enter any value.

On calculation, the selected stream and property values are assigned asmissing values.

Note that the selections associated with a particular property class persist sothat when the user changes the property class selection, the calculation optionand its associated parameters last entered by the user for that property aredisplayed.When the user clicks on the Calculate button the program will calculate theSDs only for the Stream and Property Class that are currently selected and inview.Note that even after SDs have been automatically calculated, the user is stillable to go to an individual experimental data point in the Stream Data windowand change the SD for that data point. However any changes to SD values inthe Stream Data window will not be reflected in the Automatic SD Calculationwindow.It is up to the user to decide which standard deviation calculation method bestsuits the data.

Click Close to exit the Automatic SD Calculation window.

IMPORTANTNOTE

All data points MUST have a standard deviation value associated with them or else beset to Missing. This includes the TPH water and % solids, as well as the TPH solids,overall assays and size-by-assay data.The water balance is performed on TPH water, using the experimental % solids andTPH solids to calculate the experimental values. The balanced water flow rates are

determined from the % solids values.

SelectingStreams andEquipment

The Mass Balance feature of JKSimMet allows the user to select the streams andequipment items to include in the balance, as well as selecting which parameters canbe adjusted during the mass balancing process.Open the M ass Bal ance window by either pressing the Run Balance button on thetoolbar or by selecting Run Mass Balance from the Flowsheet drop-down menu. Once




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Selecting Data

this window is open, the user enters mass balance mode where no alterations to theflowsheet, streams or equipment can be made.

For this tutorial, mass balancing will be performed on the Learner Project, which hasalready been established and has been used for examples throughout this help file.The Learner Project file will have been supplied with your copy of the software.You can either follow along using this project, or alternatively if you have your ownproject with data, you can use the instructions as a general guide for mass balancingyour own flowsheet. See the section Create a New Project if you are unfamiliar withthe steps necessary to establish a project and you wish to create and then load datainto your own project. .Step 1 Load the Learner Project and select the Learner Flowsheet from the drop-

down list. If necessary, resize the flowsheet window to view the entireflowsheet. Ensure that the flowsheet is locked.

Step 2 Click on the Mass Balance button ( ) on the toolbar to bring the MassBalance window into view.

The left-hand panel of the Mass Balance window is where you set up the conditionsfor the mass balance that is to be carried out on your circuit. Here you select whichstreams and equipment items are to be included in the balance and you specify thecomponents including the size fractions and elements (mineral or chemical assays)that will be available for the balancing procedure.At the bottom of this panel you have the Control fields where you select what is to beincluded in the current balance and you specify values for the parameters that controlthe balance.The right-hand panel contains all of the results for your balance. When the massbalance is under way, you can observe its progress in the top two fields which showthe number of iterations that have so far taken place plus the current convergencevalue. The balance will be complete when the convergence value goes below theconvergence target that has been set in the Controls section.




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As is also the case for the Simulation and Model-Fitting modules, you may select asingle unit or a cluster of units from your flowsheet on which to perform a massbalance. This allows you to check small parts of a circuit for data consistency.

Step 1 Open the Mass Balance window ( ).Step 2 When you first open this window there will be a default Mass Balance Case

called MBal Case 1 in the selection drop-down at top left. There is a

Rename button that can be used to change the name of this Test Case tosomething more appropriate, if required. The New button can be used tocreate further Test Cases with different balance settings, which can then besaved and returned to later. There is also a Delete button provided so thatyou can eliminate any test cases that you have previously created but nolonger need.

The default selection for a new Mass Balance Case is for all of the units and streams tobe selected. Therefore the first stage in defining a Mass Balance Case is to examine thelist of equipment and streams and to decide which items are to be included in yourbalance.The next step is to remove any unwanted items by clicking on the adjacent check-boxto remove the tick. Select All and Select None buttons have also been provided tospeed up the process of balance configuration. In this case we will be balancing withall equipment units selected and all streams - apart from the water streams, which wewill add later.Note that if you de-select an item of equipment, the input streams for that equipmentwill no longer be balanced with the output streams, even if all of these streams remainselected on the Streams list.

If only equipment has been selected, it is not necessary to specify which streams toinclude in the mass balance. The mass balancer will automatically determine whichstreams should be included in the mass balance based on which equipment has beenselected.

Designation In the streams list there is a second column labelled "Designation". There are threesettings possible for the designation field, Major, Minor and Missing.This distinction between the different categories of streams enables selected streams tobe balanced or fixed at various times throughout the mass balancing process.




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A Major stream refers to one that can be balanced initially, for example theplant feed, final concentrate and final tail streams, and then fixed with theintermediate streams subsequently adjusted within the circuit.A Minor stream is an intermediate stream that may have some datamissing, restricting its usefulness in the balance.A Missing stream is defined as a stream that is missing all data. Thesestreams can be omitted from the balance until such a stage that there isenough information in the other streams to allow a mass balance to besuccessful.

Note that streams belonging to the different categories are not treated differently inthe balancing process. The categories are simply groups to which you can assign eachof your streams and then later you can elect to keep their values fixed or adjust themduring the next run of the balancing process.There is visual feedback provided regarding the groups to which each stream has beenassigned to. When you have the Mass Balance window open, you will notice that allthe selected streams of the flowsheet are displayed in bold black, which is the defaultfor stream designated as Major. If you change a stream's designation, the colour willchange to bold blue for Minor or bold red for Missing. These are just the defaultcolours, so they can be changed to the users preference if required. This is done via theDefault Stream Properties item on the Edit menu. These visual cues can help usersidentify the equipment and streams that have been selected, which can be especiallyuseful when dealing with complex flowsheets.

By default, all streams are initially assigned to the Major group. As another exampleof where you might make use of this facility, you may have a complex circuit and wishto adjust values for a sub-section of the circuit only. In this case you could assign all ofthe streams that make up this sub-section, to one of the other two groups. You wouldthen find that during the next run of the balance, all plant values would remain fixedat their previous balanced levels and it would be only the stream values for therequired sub-section of plant that would be adjusted as a result of this balance.




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Selecting Components

Note that the group assignation for each stream can be changed in the above frame. Todo this you just click on the required stream and then click the appropriate button atthe bottom of the list. Note that this assignation can also be changed using the drop-down list in the top right corner of the Stream Data window.

Step 3 In the Equipment / Streams frame, select all of the equipment items and allstreams with the exception of the two water addition streams. Leave all thestreams with the default designation of Major.

Note that on the flowsheet there is also visual feedback for the selection of equipmentitems, with those that have been selected being highlighted in bold blue.

It is possible also to select and deselect items (both streams and equipment) for yourbalance list, by clicking on these items on the flowsheet. You will see that the boldhighlighting will disappear to indicate de-selection. Clicking on them performs atoggle action - so clicking again will re-instate the selection.

Step 4 Now examine your Mass Balance Case and ensure that:all equipment units are selectedall streams are selected except for the water addition streams (BMWater & Cyc Water)all streams are designated "Major"

Because each Mass Balance Case has a name, you may set up several different ones toexamine different sections of a circuit. You can select all streams and equipment unitson your flowsheet, a single unit (together with its input and output streams) or else aselection of units and their associated streams. Note that they need to be contiguous

i.e. with streams connecting all of the selected units, to ensure that your selected sub-circuit does get balanced as a single circuit.In V5 all stream data were stored in equipment ports. To balance a subset of the data,you needed to choose both equipment and ports (streams). In V6, the stream data arenow stored within the streams themselves, but the same rule still applies for thebalancing of data sub-sets.




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Solution Controls

The mass balancing module can perform mass balances based on two types of data;namely elements and size distributions. These can be used individually or together toachieve the balance. If you also have assays of the size fractions, then you can add anextra dimension to your balance by allowing the size by element data to be used in thebalancing process as well.Mass balancing therefore contrasts with the model-fitting and simulation modes in

JKSimMet since in the latter modes it is only ever the size distribution data that are

used.In the next frame of the Mass Balance window, you can select the elements and sizesthat are to be used for the current mass balance case.

In the left hand section of this frame is the list of elements that have been enteredduring the project configuration stage. At the end of this list you will see there is anelement called Remainder that has been added to allow for any mass not accountedfor by the sum of the defined elements. All of the elements will initially be selected bydefault. If for some reason you wish to leave out one of these elements from thebalancing process, then you can simply uncheck the relevant check-box.

To the right of the elements selection area is a similar list for sizes, which shows all thesizes (in mm) that have been defined for this project. Again they will be all selected bydefault to begin with. In this case we will de-select all of the sizes since we will bebalancing on the assays alone.

Again, with both of these lists, there is a check-box provided at the top to allow you toselect or deselect all of them. This can be useful for speeding up the process ofdefining the suite of components that are to be balanced for the balance case inquestion.

Step 1 Ensure that all of the elements have been selected, but that none of thesizes are selected.

To the right of the elements & sizes selection frame is another frame with the heading"Adjust Streams". Here there are check boxes for the 3 categories of streams. All threeof these boxes will be checked by default. In this case we want the balance to makeadjustments to all streams in the circuit and in any case, all the streams in this circuithave been left in the Major category. This means that as long as the Major check-box isticked, all streams will get adjusted. However, there is no problem with leaving ticksin the other two categories as well.

Step 2 Leave ticks in all three category check-boxes in the "Adjust Streams" frame.




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To perform a mass balance the next step is to set the controls for this balance. In theControl frame there are various drop-down selection fields where you have the abilityto choose the parameters that are to be involved in this balance. In addition, you cancontrol the way the balance begins and concludes.Step 1 It is wise to initially keep a balance as simple as possible, so for the first run

through of the current balance we will set the TPH Solids to "Adjust" andlikewise for Elements, but we will leave all the other parameter fields set to

"Unused".

Click in the white cells in the Select? column to change the settings on eachparameter. There are a total of four settings that parameters may be set to, althoughnot each setting will be available for each parameter.

Adjust The experimental values associated with this measurement are included inthe mass balance sum of squares calculation and a mass balance adjusted data value iscalculated by the mass balance which is consistent with all other data in the system.Fixed The previously calculated balance values are held at their current value andare used in the calculation of other parameters selected as Adjust. This is used, forexample, to keep flow rates fixed when adjusting size so as to stabilise theconvergence.

Influence The experimental values associated with this measurement influence theoutcome of the mass balance (i.e. are included in the sum of squares calculation) butare not returned as an adjusted consistent set of values after mass balancing. Thismeans the mass balanced values for this parameter are left as zeros after balancing.The exception to this is in the case of % Solids, where values are returned during massbalancing. The objective of the Influence option is to allow the experimental data forthe parameter concerned to contribute to a higher level balance, even though the datafor this parameter are too sparse to allow the generation of its own set of massbalanced values.Unused The experimental values associated with this measurement are not includedin the mass balance sum of squares calculation and its mass balanced value is notchanged.

If only certain elements or size fractions are to be sent to the balance, use the tick boxesprovided to select the required parameter. These will then be used according to theoptions set in the main drop-down boxes. If an inconsistent selection is made, an errormessage will be displayed when the mass balance is run.For balancing complex circuits, it is suggested to select all 'Major' streams in the firstinstance and balance (adjust) them. Once the user is happy with this balance, set theMajor streams to 'Fixed' and adjust the minor streams, then the missing streams.In this way certain parts of the circuit can be balanced and fixed to continue balancingother parts of the circuit.




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Running the Mass Balance

This is the simplest step. Once the components have been specified, the desiredequipment and streams selected and the balance controls set, the mass balancing canbegin.Click on the Start button which is located at the bottom right of the Mass Balancewindow.

The mass balancing program will run and when the balance has completed, the resultswill be summarised in the right hand panel of the Mass Balance window.In the Result s frame at the top of this panel is the actual convergence value. This is thevalue that is compared with the required Convergence value selected by the user inthe bottom drop-down of the left-hand panel. The program considers the balance iscomplete once the actual value goes below this user specified convergence setting.The right-hand field of the Result s frame is the number of iterations that the balancehas gone through before arriving at the indicated convergence value.




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The frame below the Results frame is labelled "RMS Errors" - standing for root meansquare errors. In this frame you can gauge how well the data has balanced comparedto the previous balance. Smaller values of the RMS errors will indicate that a betterbalance has been achieved.

The RMS errors listed here are divided into the various areas of balancing, in line withthe areas that can be selected from within the Controls frame:

The Totals refers to the solids mass flows alone - it does not include water.Flows refers to the volume flows in the various streams, consisting of thevolumes of both the water and solids in all streams, including slurry streamsand water feeder flows. Note that balancing on the volume flows only occurswhen TPH water has been selected as one parameters to be adjusted.Assays will have an RMS error displayed if you have selected "Adjust" in theElements drop-down from the Control frame.Sizes will have an RMS error when Sizes has been selected for adjustmentfrom the Control frame.Likewise Assay by Size will have an RMS error value if Size by Element hasbeen selected for adjustment.

Below the listing of the RMS error values is the Parity Graph which allows an easyvisual assessment of how good the balance has been. This graph is plotting theexperimental values against the balanced values, so for a perfect balance, with noadjustments needed, all the points would be sitting on the 45 degree line. The distancethat points are away from the 45 degree or parity line is a measure of how muchadjustment was needed for these parameters to achieve the balance.

GeneralisedStrategy forMass Balancing

If you have assays on most or all streams, the normal sequence for mass balancingwould be to initially set TPH Solids and Elements to "Adjust" with all otherparameters set to "Unused". Once a balance has been achieved using these parameters,you would next set both of them to "Fixed" and now set the TPH Water to "Adjust"and the % Solids to "Influence". Then after this balance has run, the next step would be(if you have sizing data), to set both water parameters to "Fixed" and set Sizes to"Adjust". Finally, if you also have Size by Element data you would set Sizes to "Fixed"and the Size by Element parameter to "Adjust".If you are working with Sizing data only and do not have any assays, you would startthe process with TPH Solids and Sizes set to "Adjust". After this balance runs, you




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would set these two parameters to "Fixed" and set the TPH Water to "Adjust" and the% Solids to "Influence".

The next step in the Learner Project is to make the TPH Solids and the Elements fixedand to balance on water, making the TPH Water set to "Adjust" and the % Solids set to"Influence. After you change these balancing settings, you should click on the Clearbutton prior to re-running the balance. The Mass Balance window will then look likethe one below.

Pressing the Start button will then produce a balance similar to the one shown below.

Since in this case we also have sizing data, the next step is to balance on these data. Toset this up you need to click in the check-box labelled "Select/Deselect All" above theSizes list in the Elements / Sizes frame. This will select all of the sizes that are presentin the size analysis for the samples in this circuit. You then need to uncheck the 13.2mm size, since this contains a value of zero for the experimental retained mass for allthree products and this will cause problems for the balancing routine.




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The Mass Balance Engine

In order to run the balance on the sizing data, you need to also switch the TPH Waterand % Solids back to "Unused". When the balance runs, you should see the windowlooking similar to the screen grab below.

In this case we do not have any size by assay data, so at this point the balancingprocess is complete. If you did have size by assay data, the next step would be to setthe Sizes back to "Fixed" and then the Size by Element to "Adjust".

The next step is to look critically at the balance results and there are various ways ofdoing this. The next topic discusses the options here and what you need to look for inexamining the balance results.

The Mass Balance Results section reports the value at which convergence occurred aswell as the number of iterations that were required to reach that value.In most cases of mass balancing, there will be numerous missing data. The idealscenario of the mass-balance algorithm is that the user may specify all their data, pressa button and have the calculated values returned in seconds. Unfortunately thenumber of missing data points generally prevents the algorithm from being quite thataccommodating. Therefore, mass-balancing needs to occur using a hierarchicalapproach; that is by first mass-balancing flows, then sizes and total assays, then assayswithin size-classes. This hierarchical approach was discussed previously . The keyinterface is therefore the Control frame in the Mass Balance window you specifywhich levels are to be mass-balanced.The mass balancing algorithm in JKSimMet allows high level variables to be calculatedusing lower level results without actually estimating the low-level values. Forexample, it is possible to use size information to estimate flows without actuallyhaving to estimate the calculated values of size. Data-checking is performed only forthose variables that are being mass-balanced (not the values of lower level values usedto influence the mass-balance). However one needs to bear in mind that unrealisticlow-level standard deviations will cause unreasonable weighting to the lower levels

Data Checking Data-checking is required to ensure that values are realistic. The data-checking occursfor all levels of variables to be calculated, and assumes that higher level values have




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reasonably small standard deviations of the calculated values. In other words data-checking treats upper level values as if they are fixed.Data-checking is used to check the reliability of the data. In particular, if standarddeviation values are too small, the algorithm will have difficulty converging.The data-checking algorithm works by considering the difference between flow (ofany property) going into and out of a unit. The difference should be consistent withthe standard deviations. For example, for a unit having one stream in and twostreams out, the total flow values may be represented by f1, f2 and f3 withcorresponding variance values v1, v2 and v3. The flow difference is:

The expected variance of this difference value is given by:

Thus the statistic:

provides an estimate of the reliability of these data. The expected value of r is 1. Thisr value is referred to as the RMS (root mean square) value.An RMS value greater than 3 usually indicates that the data contains an outlier or thatthe standard deviations are too small.

DataMismatchesDuring MassBalancing

Once the mass balance Start button has been selected, the data checking algorithmbegins. All units are checked, and if any unit has an RMS value greater than 3, thefollowing window is displayed:

This indicates the units where the RMS value is greater than 3, along with thecomponent that is responsible for the mismatch. The value calculated for the Currentmismatch is the RMS value. In the above screen grab, the Unit name is a system




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generated name which will be produced under circumstance where you have chosenstreams that are the input and outputs from a section of the circuit but withoutincluding intermediate streams.The details of the mismatch can then be seen by clicking on the button in the Detailscolumn, which brings up a dialogue box, as follows:

This dialogue box gives the feed and product streams for the particular unit, alongwith the experimental value (for the example case, experimental TPH water values),the current standard deviation, and the suggested standard deviation to remove themismatch.The method for arriving at the suggested value is as follows:

The user is given the option of accepting the suggested SD, by clicking OK, in whichcase the mismatch value in the data checking information box will be updated whenthe data mismatch box is closed. The user is also given the option of specifying a newSD. All information in the data checking information box needs to be updated afterthe data mismatch box is closed.If the user clicks Cancel, the current SD remains the same, and when the datamismatch box is closed, the current r value is calculated. The user can then choose tocontinue mass balancing, or cancel the mass balance process.

Using the RMSErrors

The Error Analysis section of the Control/Run window gives an indication of thequality of the mass balanced data through the use of RMS (root mean square) errors.As its name suggests, the root mean square error is the square root of the residualmean square error, which is the error associated with fitting the mass balanced data tothe experimental data.A root mean square error of unity means that all data points were estimated witherrors similar to the measured standard deviations of the experimental data points.Therefore, when performing a mass balance, an RMS Total value of somewherebetween 0.5 and 2.0 indicates that the experimental data is good and that the correctstandard deviations have been used.Along with the RMS Total value it is possible to check the error values (i.e. (observed predicted)/sd) on the other parameters used in the mass balance. Click the Detailbutton next to each parameter to see information about that parameter.




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Checking the Balance

This window, for instance, would indicate that particular problems exist with the CycU/F and BM Product streams.Identifying particularly problematic streams allows the user to immediately focus onwhich data values may be less accurate than indicated by their standard deviation. Itmay be necessary to change standard deviation values to improve the mass balance.

There are various ways in which the user can assess the results:

The Overview window is probably the most useful way to check data andresults. It also allows recovery of any component to be displayed for allstreams. See Learning Overview for details of the Overview facility.In the RMS Errors frame of the Mass Balance window, the magnitude of theaccumulated error values under the various parameter categories, are allindicators of how close the original experimental data were to being alreadybalanced. Smaller values of the RMS Errors indicate less adjustment wasrequired to produce the balance. Moreover, in the case of these fields,comparisons between successive mass balances can be made. If these valuesare smaller than they were in the previous run, then the balance is gettingbetter. If they are getting larger, then the adjustments you have made aretaking this balance in the wrong direction.You can also judge the relative success of mass balancing by looking at theindividual stream data windows. Examine the values in the Error column.Again, smaller error values indicate less adjustment has been required.The parity graph on the Mass Balance window allows for a good visualassessment of the balance. The further the data points are away from the 45degree line, the greater was the adjustment needed to balance the parametersin question. The TPH Solids data usually consist of numbers that are very largecompared to the assay data and when the two are plotted together on thisgraph, the assay data will lie in a clump near the origin, which is not veryuseful for assessment. To overcome this problem, you can temporarily switchthe TPH Solids to "Unused", while leaving Elements on "Adjust". You will alsohave to de-select "Remainder" from the elements list, since this will be arelatively large number also, possibly approaching 100. After you have donethis, you should get a much better picture of how well the elemental data havebalanced. It should be noted though that you cannot actually carry out a




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Presentation of Mass Balance Results

Configurable Stream Overview

balance with the TPH Solids set to "Unused". Note too that you can hover themouse over individual data points to identify them, as is illustrated in thescreen grab below. Don't be too impatient here though - you do need to havethe mouse positioned over the point you wish to identify for between 1 and 2seconds before the data will appear.

The Configurable Graphing facility allows the user to plot experimental andbalanced size distribution data on the same screen. Further description of thiscan be found in the section Plotting Size Data Graphs .

If mass balancing a large and complex circuit is proving difficult, a usefultechnique for tracing the source of the problem is to dissect the circuit intosmaller chunks for balancing. The Mass Balancing module allows balances tobe carried out on a single unit or on a small set of units, isolated from the maincircuit by means of the Select List facility. This allows you to put the test dataunder a microscope.

If circuit conditions were changing as you did your test work, you may find that theunstable sections of the plant will have yielded unusable results. As a generalprincipal, a good balance depends on having steady state conditions and varying

conditions will usually produce nonsense.

There are two main ways to present the results of mass balancing:viewing on screen via the Configurable Stream Overview window andprinting or exporting via the Reports function.

We shall deal with these in turn. For mass balanced data, graph plotting is limited toGSIM format.

UsingConfigurableStreamOverview toView BalanceResults

The Configurable Stream Overview window gives you a powerful means ofsummarising your data and checking it for adjustment problems. There are somebuilt-in overviews that will already be present when you first open the ConfigurableStream Overview. This includes one that contains Experimental, Balanced and SDdata. This one is all you would normally need for assessing the results of a massbalancing procedure. However, as implied by the name, this overview window can beconfigured by the user and you can create your own overviews with additional datadisplayed or with just a subset of the streams in your flowsheet displayed.




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Reports Function and Printing

The best way to use the overview feature is to compare experimental and balancedvalues for each assay (or size fraction) across the complete circuit. This will give a veryuseful picture of the accuracy of the data and the mass balance.

Step 1 Left-click on the Configurable Stream Overview button ( ) on the main JKSimMet toolbar. This brings the Configurable Stream Overview windowinto view.

Step 2 Select From the Select List at top left, select the pre-configured overviewthe existing Overview that is labelled "Stream Data (Exp, Bal & SD)".

Note that from within this window, you can elect to change the way you view thesizing data. The drop-down at top right contains the sizing data options of %Retained, Cum % Retained and Cum % Passing. You can also gain access to the SDCalculation window from here via the button next to this drop-down. This is providedhere since after examining the most recent balance data, it will often be the case thatyou may need to modify certain SD values before re-running the balance.

Printing MassBalance Results

Normally for reporting the results of a mass balancing procedure you would mostlikely want to show the experimental, balanced and SD values for each type of datainvolved and for each of the streams included in the balance.

PrintingIndividual PortData Windows

One way to do this is via the individual stream data windows. To open a stream datawindow, just double-click on the required stream in your flowsheet. These windowsall have SD and Mbal data as pre-configured display columns. You cannot directlyprint these windows, but you can copy the grid from them into Excel. To do this youpress the Copy button, located top centre of the Stream Data window. As it is only theactive tab that gets copied, you need to ensure that you have the appropriate tabselected for the type of data that you are intending to report on.This method may occasionally be useful if you are wishing to report the data for aparticular stream, but in general one of the other two methods below is probably more

practical.Printing theConfigurableStreamOverview

Another alternative is that once you have the Configurable Stream Overview set upaccording to your requirements, you can either print it directly or else export it toExcel for further manipulation before printing.When you use the Print function directly, the options for manipulating the format ofyour report are somewhat limited and it is likely to span across several pages. In mostcases it is probably better to use the Export function and then print a report fromExcel.There are two Export buttons - one is for exporting just the overview that is currentlydisplayed, while the other will export all of the overviews that have been set up for




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and balanced data. A drop-down is provided to allow you to select the format of thesizing data. This can be either cumulative % passing, % retained or cumulative %retained. The data can be plotted for a number of streams on the same chart, to allowcomparisons.An example of the graphing facility with the required data types in the process ofbeing selected from the drop-down, is shown in the screen grab below.

Once the streams to be displayed have been selected and the data format and datatypes selected, you simply click on the Graph tab and the required chart will appear inplace of the graph configuration table. The screen grab below shows a typical examplein which the user has chosen to show the circuit feed from the Learner Flowsheet plusthe two products from the cyclone.




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Problems Related to Mass Balancing and Possible Solutions

It is also possible to make changes to the way this graph appears, using the Formatbutton near the top right of this window. If you click on this button, a new windowwill open where you can make a number of changes to the chart formatting. Anexample is shown below.

Any graph that you generate here can be easily inserted into an Excel or MS Wordreport using the Copy button at top right.




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The Middlings Problem

There are, of course, many problems that may be encountered during mass balancing.It is possible however, to point out some of the more common problems and adescription of these has been set out in the remainder of this section. While the listbelow is not comprehensive, we hope that it will alert you to some of the moresignificant pitfalls.

Errors, Warnings, Faults

Some problems detected by JKSimMet can produce error messages. In version 5 theerrors had error numbers and were referenced via a key that could be found in thehelp file. In version 6 it is no longer necessary to refer to a key since the errorexplanations now appear in the error message window.

GraphicalAnalysis

The graphing capability of JKSimMet is the most powerful way to examine your datafit. Discontinuities in size data highlight poor data or a change in size measurementtechnique. Graphical analysis also highlights any bias in the data fit.

Different SizingTechniques

Be very wary of changes in size measurement technique e.g. from screens to aCyclosizer.

Different AssayTechniques

Where assay techniques change between stream samples, as they sometimes do fordifferent assay ranges, there may be inherent biases within the assay techniques. Thesewill lead to biases within the mass balance.

Skill versusPractice Mass Balancing is not a cut and dried procedure. The only way to acquire a useful skilllevel is to practise on a wide range of real data. JKSimMet offers a user-friendlyenvironment for what are really very complex and powerful mathematical techniques.

Data Note that it is necessary to have enough feed and product data to achieve a usefulmass balance. This is very important. Generally you need to have redundant data andthe more of this you have, the better. If you have no redundant data at all, then themass balancing exercise reduces to just a calculation i.e. there is only one solution tothe balance. In this case the internal checking that comes from having these extra data(and is also one of the major advantages of mass balancing), will no longer beoperating. Without redundant data, there could well be a very large error in one ofyour data points and you would have no way of ever being alerted to it.

Common MassBalancingPitfalls

There are a couple of simple traps which can appear in many guises. If you becomeaware of these now you may recognise them more easily when you encounter them inthe future. These are discussed in the two sub-topics that follow.

Below we have a single unit flow diagram for a separation node where there is amiddlings stream of assay m.

In this situation there are not enough assays to go around.However, if we have two assays in each stream, we would write them out as simpleequations and solve for two unknowns. However, if m really is a middlings stream, itwill be close to a in composition and very often recycled back to it.

In this case, no matter how accurately we can sample and assay the streams, we canonly find out:

the ratio between flows b & c (if m goes elsewhere)or




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The Infinite Division Problem

Metallurgical Accounting

the flows in b and c if m is recycled.The actual flow rate in m can be anywhere between zero and infinity. However, thereis a straightforward solution. Measure (or estimate) the flow rate in stream m andinput this flow rate as data.The mass balancing module allows you to do this.

The InfiniteDivisionProblem

If one wishes to extract maximum information from a survey, it is not unusual to assayon a two (or even three) dimensional matrix, for example, assay by size or assay bysize by specific gravity. This subdivides the stream into even smaller sub-groups. Eachsub-group has an extra step of processing and an increased relative error. Hence, wetend towards trying to solve for (0 - 0) / (0 - 0). This is not a useful numerical exercise.

The solution to this problem is straightforward, however.You should first use the total assays where there are large differences, to calculate themass balance flow rate solutions. Once you have these flow rates, fix them by selecting"Fixed" from the TPH Solids drop-down (in the balance controls frame) and only thenadd in all of the smaller fraction assays to the mass balancing problem.Once the flow rates have been defined, the mass balancing module will be more easilyable to allocate the minimum adjustments required to make all of the fractional assaysconsistent.

Use inMetallurgicalAccounting

The day to day data collected from a mineral processing plant are rarely consistentand will almost always contain redundant information. In general, any two methodsof calculation will yield different results. The challenge for metallurgical accounting isto produce adjusted data that are both self-consistent and as accurate a representationof plant performance as possible.

Consider a typical base metal concentrator with several products from several circuits,

At each point marked , we have Au, Cu, Fe, Pb and Zn Assays. For the feed, we haveweightometer readings and for the concentrates we have load out weights withstockpile surveys.If we select an accounting period that is large compared with the circuit residence

time, we can carry out a mass balance over this complete data set. If large adjustmentsare required, these may be an indication of problems in either sampling or assaytechniques. In this case you may need to select smaller circuits for mass balancing inorder to isolate and identify these problems.Once a consistent set of adjusted data is produced for each accounting period, thesums of these sets will also be consistent.If assays and flow rates are available on a short time scale, e.g. several times per shift,these data can be balanced for each time period, printed to a file or exported to mostWindows spreadsheet or word processing packages using copy and paste.




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References

JKMetAccount For users with a serious interest in metallurgical accounting, the JKMetAccountprogram was created to enable the Metallurgist or Plant Manager to track theperformance of a mineral processing plant over time.Changes to a plant flowsheet,which can so often cause problems for a spreadsheet based system, are easilyaccommodated by the JKMetAccount software.The major strength of this program comes from harnessing the power of the JKMBalmass balance engine (and more recently a model based mass balance engine), within a

rigorous data management environment that is accessed via a user friendly graphicalinterface. JKMetaccount also still provides for very flexible reporting, as it utilises theformatting power of Excel within its report production module.With its rigorous data management advantage over a spreadsheet system, we believethat JKMetAccount will in time become regarded as an indispensable tool for modernmineral processing plants.

At the end of 2006, all rights to the JKMetAccount program were sold to the largeBrisbane based software development company Mincom, whom were consideredbetter able to handle the further development and marketing of this product. Mincomhave since been taken over (in July 2011) by ABB and are now part of the ABB ownedcompany Ventyx.If you are interested in finding out more about JKMetAccount, further information canbe obtained either from the JKTech website(http://www.jktech.com.au/commercialisation-case-studies ) or directly from Ventyx,whose contact details can be obtained from their website at http://www.ventyx.com/ .

LYNCH, A.J., 1977. Mineral Crushing and Grinding Circuits, (Elsevier, Amsterdam),Chapter 7.

LYMAN, G.J., 1986. Application of Gy's sampling theory to coal, International Journalof Mineral Processing, Vol 17:1-22.

GY, P.M., 1982. Sampling of particulate materials: theory and practice, 2nd Ed,(Elsevier, Amsterdam), p 431.

MORRISON, R.D., 1976. A two stage least squares technique for the general materialbalance problem, JKMRC Internal Report No 61 (unpublished)


Documents

6.Mass Balancing