Partition (Density, Pivot): Difference between revisions
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== Model theory == | == Model theory == | ||
[[File:PartitionPivot2.png|thumb| | [[File:PartitionPivot2.png|thumb|500px|Figure 1. Pivot model partition to sinks. Each coloured line represents a particle class with the geometric mean size indicated in the series label. The pivot point is evident at a density of 2.0 t/m<sup>3</sup>.]] | ||
[[File:PartitionPivot3.png|thumb| | [[File:PartitionPivot3.png|thumb|500px|Figure 2. Pivot model partition to sinks, using the same model parameters as Figure 1. Each coloured line represents a density class with the specific gravity indicated in the series label.]] | ||
Scott and Napier-Munn (1992) derived an empirical equation that describes the partition of particles of a given size and density to the sinks stream of a dense medium cyclone.{{Scott and Napier-Munn (1992)}} | Scott and Napier-Munn (1992) derived an empirical equation that describes the partition of particles of a given size and density to the sinks stream of a dense medium cyclone.{{Scott and Napier-Munn (1992)}} | ||
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Several metrics are provided to characterise the Pivot model partition curve. | Several metrics are provided to characterise the Pivot model partition curve. | ||
The <math>d_{50}</math>, also known as the | The <math>d_{50}</math>, also known as the ''cut or separation size'', is defined as the size of a particle of a given density which has an even (50%) chance of appearing in either the sinks or floats stream. The <math>d _{50}</math> size is obtained by rearranging the Pivot model: | ||
:<math>d_{50} = \left ( \dfrac{\ln 3}{k} \dfrac{(\rho_j - \rho_{\rm p})}{\ln ({Y_{\rm p}}^{-1} - 1)} \right ) ^ {\frac{1}{n}}</math> | :<math>d_{50} = \left ( \dfrac{\ln 3}{k} \dfrac{(\rho_j - \rho_{\rm p})}{\ln ({Y_{\rm p}}^{-1} - 1)} \right ) ^ {\frac{1}{n}}</math> | ||
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which is only defined when <math>(\rho_j > \rho_{\rm p}, Y_{\rm p} \leq 0.5)</math> or <math>(\rho_j < \rho_{\rm p}, Y_{\rm p} \geq 0.5)</math> | which is only defined when <math>(\rho_j > \rho_{\rm p}, Y_{\rm p} \leq 0.5)</math> or <math>(\rho_j < \rho_{\rm p}, Y_{\rm p} \geq 0.5)</math> | ||
The <math>\rho_{50}</math>, also known as the | The <math>\rho_{50}</math>, also known as the ''cut or separation density'', is defined as the density of a particle of a given size which has an even (50%) chance of appearing in either the sinks or floats stream. The <math>\rho _{50}</math> density is obtained by rearranging the Pivot model: | ||
:<math>\rho_{50} = \rho_{\rm p} + \dfrac{k \bar {d_i}^n}{\ln 3} \ln ( {Y_{\rm p}}^{-1} - 1)</math> | :<math>\rho_{50} = \rho_{\rm p} + \dfrac{k \bar {d_i}^n}{\ln 3} \ln ( {Y_{\rm p}}^{-1} - 1)</math> | ||
The | The ''Ecart Probable'', or <math>E_{\rm p}</math>, is a measure of the deviation of a partition curve from a perfect separation, and is typically defined for density separations as: | ||
:<math>E_{\rm p} = \dfrac{\rho_{75} - \rho_{25}}{2}</math> | :<math>E_{\rm p} = \dfrac{\rho_{75} - \rho_{25}}{2}</math> | ||
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where <math>^*</math> indicates optional results returned if <math>\mathit{returnPartitionMetrics} = \text{True}</math> | where: | ||
* <math>n</math> is the number of size intervals | |||
* <math>m</math> is the number of density classes | |||
* <math>^*</math> indicates optional results returned if <math>\mathit{returnPartitionMetrics} = \text{True}</math> | |||
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|EpExponent | |EpExponent | ||
|Input | |Input | ||
| | |Exponent of the size equation. | ||
{{SysCAD (Text, Dense Medium, Liquids)|method=0}} | {{SysCAD (Text, Dense Medium, Liquids)|method=0}} |
Revision as of 03:53, 16 January 2024
Description
This article describes the Pivot model equation for the partition of particles by size and density in a dense medium cyclone.[1]
The Pivot model may also be applicable to other dense medium and gravity concentration processes, based indirectly on Rao et al.'s (2003) derivation of the Pivot model from the Stochastic model and the presented data analyses.[2]
Model theory
Scott and Napier-Munn (1992) derived an empirical equation that describes the partition of particles of a given size and density to the sinks stream of a dense medium cyclone.[1]
Scott and Napier-Munn's approach is based on the observed pivot phenomenon where the partition curves of particles of all sizes appear to 'pivot' at a unique density value, as illustrated in Figure 1.
The proposed Pivot model equation is:
where:
- is the index of the size interval, , is the number of size intervals
- is the index of the density class, , is the number of density classes
- is the fraction of the mass of particles in size class and density class which are partitioned (recovered) to the sinks stream (frac)
- is the pivot density (t/m3)
- is the particle density (t/m3)
- is the partition to sinks at the pivot density (frac)
- is the Ecart Probable size coefficient
- is the Ecart Probable size exponent
- is the geometric mean size of particles in size interval (mm)
Partition metrics
Several metrics are provided to characterise the Pivot model partition curve.
The , also known as the cut or separation size, is defined as the size of a particle of a given density which has an even (50%) chance of appearing in either the sinks or floats stream. The size is obtained by rearranging the Pivot model:
which is only defined when or
The , also known as the cut or separation density, is defined as the density of a particle of a given size which has an even (50%) chance of appearing in either the sinks or floats stream. The density is obtained by rearranging the Pivot model:
The Ecart Probable, or , is a measure of the deviation of a partition curve from a perfect separation, and is typically defined for density separations as:
where and are the densities of particles which have a 75% and 25% probability, respectively, of appearing in the sinks stream.
Middlings
Gravity concentrators such as jigs, spirals and shaking tables produce a bed or band of partially stratified components at the point of discharge. Portions of the bed or band are then typically directed to product streams by a physical device, such as a weir, 'splitter' or 'cutter'. These devices are usually adjustable, and can be arranged to recover arbitrary fractions of the bed or band.
From a physical standpoint, adjusting the discharge device to recover more of the bed or band has the effect of recovering the both the portion from the previous position plus the portion in between the previous and new positions. As more mass is recovered by this process, the partition curve effectively 'shifts upwards'. The partition curve is thus representing the cumulative recovery of mass from all positions between the beginning of the bed/band and the discharge device position.
Mathematically, the partition curve generated by such a gravity concentration method should also be considered a cumulative recovery of mass to concentrate. When multiple product streams exist, e.g. concentrate and middlings, the partition of components to each individual product stream will be the difference between the cumulative partition curves at each product stream position.[3] That is,
where:
- is the index of the product stream, i.e. is the first concentrate stream, are subsequent lower-grade concentrate or middlings streams
- is the mass fraction of particles in the feed stream in size class and density class which are partitioned to the product stream (frac)
- is the cumulative mass fraction of particles in the feed stream in size class and density class which are partitioned to all the products streams up to and including (frac)
- The term prevents negative partition values
The cumulative recovery formulation described above is physically and mathematically distinct from staged recovery processes which apply partition curves to the unrecovered streams of previous partition steps.
Excel
The Pivot partition equation may be invoked from the Excel formula bar with the following function call:
=mdPartition_Pivot(MeanSize as Range, RD as Range, Yp as Double, rhop as Double, k as Double, n as Double, Optional returnPartitionMetrics as Bool = false)
Invoking the function with no arguments will print Help text associated with the model, including a link to this page.
The input parameters and model results are defined below in matrix notation, along with an example image showing the selection of the same cells in the Excel interface:
where:- is the number of size intervals
- is the number of density classes
- indicates optional results returned if
SysCAD
The Pivot partition is available from the MetDynamics*GravityConcentrator and MetDynamics*DenseMedium unit models.
The sections and variable names used in the SysCAD interface are described in detail in the following tables.
Note that a Con and Partition page is provided provided for each connected concentrate discharge stream of the MetDynamics*GravityConcentrator model.
MD_GravityConcentrator page
The first tab page in the access window will have this name.
Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
---|---|---|
Tag | Display | This name tag may be modified with the change tag option. |
Condition | Display | OK if no errors/warnings, otherwise lists errors/warnings. |
ConditionCount | Display | The current number of errors/warnings. If condition is OK, returns 0. |
GeneralDescription / GenDesc | Display | This is an automatically generated description for the unit. If the user has entered text in the 'EqpDesc' field on the Info tab (see below), this will be displayed here.
If this field is blank, then SysCAD will display the unit class ID. |
Requirements | ||
On | CheckBox | This enables the unit. If this box is not checked, then the MassFracToCon option appears below. |
MassFracToCon | Input | Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the concentrate stream when the model is off. |
Options | ||
ShowQFeed | CheckBox | QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream. |
SizeForPassingFracCalc | Input | Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section. |
FracForPassingSizeCalc | Input | Fraction passing for Size calculation. The fraction input here will be shown in the Stream Summary section. |
Stream Summary | ||
MassFlow / Qm | Display | The total mass flow in each stream. |
SolidMassFlow / SQm | Display | The Solids mass flow in each stream. |
LiquidMassFlow / LQm | Display | The Liquid mass flow in each stream. |
VolFlow / Qv | Display | The total Volume flow in each stream. |
Temperature / T | Display | The Temperature of each stream. |
Density / Rho | Display | The Density of each stream. |
SolidFrac / Sf | Display | The Solid Fraction in each stream. |
LiquidFrac / Lf | Display | The Liquid Fraction in each stream. |
Passing | Display | The mass fraction passing the user-specified size (in the field SizeForPassingFracCalc) in each stream. |
Passes | Display | The user-specified (in the field FracForPassesSizeCalc) fraction of material in each stream will pass this size fraction. |
Con page
The Con page is used to specify the selected model input parameters.
Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
---|---|---|
Stage | ||
On | Checkbox | This enables the stage. If off, the feed to this stage passes directly to the next stage (or tail) without partition. |
Method | Partition (User) | The partition to concentrate for each size interval is defined by the user. |
Partition (Pivot) | The partition to concentrate for each size interval is defined by the Pivot model. | |
Partition (Stochastic) | The partition to concentrate for each size interval is defined by the Stochastic model. | |
Partition (Bazin) | The partition to concentrate for each size interval is defined by the Bazin model. | |
Jig (King) | The partition to concentrate for each size interval is defined by the King jig stratification model. | |
Pivot | ||
HelpLink | Opens a link to this page using the system default web browser. Note: Internet access is required. | |
Parameters | ||
PivotPartition / Yp | Input | Partition at pivot point. |
PivotDensity / Rhop | Input | Density at pivot point. |
EpCoefficient | Input | Coefficient of the size equation. |
EpExponent | Input | Exponentof the size equation. |
Liquids | ||
LiquidsSeparMethod | Split To Con (User) | Liquids are split to concentrate by a user-defined fraction of liquids in the feed. |
Con Solids Fraction | Sufficient liquids mass is recovered to the concentrate stream to yield the user-defined concentrate solids mass fraction value (if possible). | |
Con Liquids Fraction | Sufficient liquids mass is recovered to the concentrate stream to yield the user-defined concentrate liquids mass fraction value (if possible). | |
ConSolidsFracReqd / Con.SfReqd | Input | Required value of the mass fraction of solids in the concentrate stream. Only visible if Con Solids Fraction is selected. |
ConLiquidsFracReqd / Con.LfReqd | Input | Required value of the mass fraction of liquids in the concentrate stream. Only visible if Con Liquids Fraction is selected. |
LiqSplitToCon / Con.LiqSplit | Input/Display | The fraction of feed liquids recovered to the concentrate stream. |
Partition page
The Partition page is used to specify or display the partition by species and size values.
Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
---|---|---|
Distribution | ||
Name | Display | Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream. |
IntervalCount | Display | Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream. |
SpWithPSDCount | Display | Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
|
CumulativePartition | ||
Method | Model/User | Select model-calculated or user-defined cumulative partition to separate each solids species type. |
Density | Display | Density of each solid species. |
Size | Display | Size of each interval in mesh series. |
MeanSize | Display | Geometric mean size of each interval in mesh series. |
All (All column) | Display |
|
CumulativePartition | Display |
|
All (All row, All column) | Display |
|
All (All row, per species) | Display |
|
Partition | ||
Method | Model/User | Select model-calculated or user-defined partition to separate each solids species type. |
Density | Display | Density of each solid species. |
Size | Display | Size of each interval in mesh series. |
MeanSize | Display | Geometric mean size of each interval in mesh series. |
All (All column) | Display |
|
Partition | Display |
|
All (All row, All column) | Display |
|
All (All row, per species) | Display |
|
PartitionMetrics | ||
d50 | Display | Cut size (d50) of separation of each ore species. |
Size | Display | Size of each interval in mesh series. |
MeanSize | Display | Geometric mean size of each interval in mesh series. |
CutDensity / Rho50 | Display | Cut density (Rho50) of all particles in each size interval. |
EcartProbable / Ep | Display | Ecart Probable of all particles in each size interval. |
PivotPartition / Yp | Display | Partition at pivot point. |
MD_DenseMedium page
The first tab page in the access window will have this name.
Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
---|---|---|
Tag | Display | This name tag may be modified with the change tag option. |
Condition | Display | OK if no errors/warnings, otherwise lists errors/warnings. |
ConditionCount | Display | The current number of errors/warnings. If condition is OK, returns 0. |
GeneralDescription / GenDesc | Display | This is an automatically generated description for the unit. If the user has entered text in the 'EqpDesc' field on the Info tab (see below), this will be displayed here.
If this field is blank, then SysCAD will display the unit class ID. |
Requirements | ||
On | CheckBox | This enables the unit. If this box is not checked, then the MassFracToSinks option appears below. |
MassFracToSinks | Input | Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the sinks stream when the model is off. |
Method | Partition (User) | The partition to sinks for each size interval is defined by the user. |
Partition (Pivot) | The partition to sinks for each size interval is defined by the Pivot model. | |
Partition (Stochastic) | The partition to sinks for each size interval is defined by the Stochastic model. | |
Drum (Baguley) | The Baguley dense medium drum model is used to determine the partition of solids to sinks and floats for each size interval. | |
Options | ||
ShowQFeed | CheckBox | QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream. |
ShowQSinks | CheckBox | QSinks and associated tab pages (eg Sp) will become visible, showing the properties of the sinks stream. |
ShowQFloats | CheckBox | QFloats and associated tab pages (eg Sp) will become visible, showing the properties of the floats stream. |
SizeForPassingFracCalc | Input | Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section. |
FracForPassingSizeCalc | Input | Fraction passing for Size calculation. The fraction input here will be shown in the Stream Summary section. |
Stream Summary | ||
MassFlow / Qm | Display | The total mass flow in each stream. |
SolidMassFlow / SQm | Display | The Solids mass flow in each stream. |
LiquidMassFlow / LQm | Display | The Liquid mass flow in each stream. |
VolFlow / Qv | Display | The total Volume flow in each stream. |
Temperature / T | Display | The Temperature of each stream. |
Density / Rho | Display | The Density of each stream. |
SolidFrac / Sf | Display | The Solid Fraction in each stream. |
LiquidFrac / Lf | Display | The Liquid Fraction in each stream. |
Passing | Display | The mass fraction passing the user-specified size (in the field SizeForPassingFracCalc) in each stream. |
Passes | Display | The user-specified (in the field FracForPassesSizeCalc) fraction of material in each stream will pass this size fraction. |
Pivot page
The Pivot page is used to specify the required model method and associated input parameters.
Partition page
The Partition page is used to specify or display the partition by species and size values.
Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
---|---|---|
Distribution | ||
Name | Display | Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream. |
IntervalCount | Display | Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream. |
SpWithPSDCount | Display | Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality. |
Partition | ||
Method | Model/User | Select model-calculated or user-defined partition to separate each solids species type. |
Density | Display | Density of each solid species. |
Size | Display | Size of each interval in mesh series. |
MeanSize | Display | Geometric mean size of each interval in mesh series. |
All (All column) | Display |
|
Partition | Display |
|
All (All row, All column) | Display |
|
All (All row, per species) | Display |
|
PartitionMetrics | ||
d50 | Display | Cut size (d50) of separation of each ore species. |
Size | Display | Size of each interval in mesh series. |
MeanSize | Display | Geometric mean size of each interval in mesh series. |
CutDensity / Rho50 | Display | Cut density (Rho50) of all particles in each size interval. |
EcartProbable / Ep | Display | Ecart Probable of all particles in each size interval. |
PivotPartition / Yp | Display | Partition at pivot point. |
About page
This page is provides product and licensing information about the Met Dynamics Models SysCAD Add-On.
Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
---|---|---|
About | ||
HelpLink | Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required. | |
Information | Copies Product and License information to the Windows clipboard. | |
Product | ||
Name | Display | Met Dynamics software product name |
Version | Display | Met Dynamics software product version number. |
BuildDate | Display | Build date and time of the Met Dynamics Models SysCAD Add-On. |
License | ||
File | This is used to locate a Met Dynamics software license file. | |
Location | Display | Type of Met Dynamics software license or file name and path of license file. |
SiteCode | Display | Unique machine identifier for license authorisation. |
ReqdAuth | Display | Authorisation level required, MD-SysCAD Full or MD-SysCAD Runtime. |
Status | Display | License status, LICENSE_OK indicates a valid license, other messages report licensing errors. |
IssuedTo | Display | Only visible if Met Dynamics license file is used. Name of organisation/seat the license is authorised to. |
ExpiryDate | Display | Only visible if Met Dynamics license file is used. License expiry date. |
DaysLeft | Display | Only visible if Met Dynamics license file is used. Days left before the license expires. |
Additional notes
- Solid species that do not possess a particle size distribution property are split according to the overall mass split of the default particle size distribution species selected in the SysCAD Project Configuration.
- If the default particle size distribution species is not present in the unit feed, the overall split of all other species with particle size distributions combined is used, as determined by the model.
- Gas phase species report directly to the tail/floats stream without split.
See also
References
- ↑ 1.0 1.1 Scott, I.A. and Napier-Munn, T.J., 1992. A dense medium cyclone model based on the pivot phenomenon. Transactions of the Institute of Mining and Metallurgy Section C Mineral Processing and Extractive Metallurgy, 101, pp.C61-C76.
- ↑ Rao, B.V., Kapur, P.C. and Konnur, R., 2003. Modeling the size–density partition surface of dense-medium separators. International Journal of Mineral Processing, 72(1-4), pp.443-453.
- ↑ King, R.P., Juckes, A.H. and Stirling, P.A., 1992. A quantitative model for the prediction of fine coal cleaning in a spiral concentrator. Coal preparation, 11(1-2), pp.51-66.