Partition (Size, Bazin)

Description

This article describes the Bazin et al. (2014) empirical expression for the recovery of particles to the concentrate stream of a spiral concentrator.

The form of the empirical equation may also be applicable to other gravity concentration processes exhibiting a similar reverse classification effect on coarser particles, such as shaking tables and the Falcon gravity concentrator.

Model theory

Figure 1. Bazin partitions to concentrate for three minerals with differing densities (after Bazin et al. (2014).

Bazin et al. (2014) proposed an empirical expression for the partition of particles of a given size and density to the concentrate stream of a spiral concentrator. The concentrate stream is defined here as the inner part of the spiral trough.

The expression is formulated as the product of two counter-acting mechanisms that simultaneously force heavy minerals to the centre of the trough and coarse particles to the outer edge of the trough.

The partition of particles of a given size and density to the concentrate stream, $y(d,\rho )$ (frac), is:

$y(d,\rho )=S_{\rm {c}}(d,\rho )\times P_{\rm {R}}(d,\rho )$

where:

• $S_{\rm {c}}(d,\rho )$ is the fraction of particles which should nominally report to the inner part of the trough (concentrate stream) due to gravitational forces, the classification action
• $P_{\rm {R}}(d,\rho )$ is the probability that a particle resists the inward forces and is directed towards the outer trough by the Bagnold force, the filtration action
• $d$ and $\rho$ are the diameter (mm) and density (t/m³) of the particle respectively

For series of size intervals, the $S_{\rm {c}}(d,\rho )$ and $P_{\rm {R}}(d,\rho )$ terms are computed as:

$S_{\rm {c}}(d_{i},\rho _{j})={\dfrac {\exp \left(\alpha _{\rm {S}}(\rho _{j}){\dfrac {{\bar {d}}_{i}}{d_{50;{\rm {S}}}(\rho _{j})}}\right)-1}{\exp \left(\alpha _{\rm {S}}(\rho _{j}){\dfrac {{\bar {d}}_{i}}{d_{50;{\rm {S}}}(\rho _{j})}}\right)+\exp {\big (}\alpha _{\rm {S}}(\rho _{j}){\big )}-2}}$

$P_{\rm {R}}(d_{i},\rho _{j})=R_{\rm {P}}(\rho _{j})+(1-R_{\rm {P}}(\rho _{j}))\left\{1-\left[{\dfrac {\exp \left(\alpha _{\rm {P}}(\rho _{j}){\dfrac {{\bar {d}}_{i}}{d_{50;{\rm {P}}}(\rho _{j})}}\right)-1}{\exp \left(\alpha _{\rm {P}}(\rho _{j}){\dfrac {{\bar {d}}_{i}}{d_{50;{\rm {P}}}(\rho _{j})}}\right)+\exp {\big (}\alpha _{\rm {P}}(\rho _{j}){\big )}-2}}\right]\right\}$

$Y_{ij}=S_{\rm {c}}(d_{i},\rho _{j})\times P_{\rm {R}}(d_{i},\rho _{j})$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $j$ is the index of the density class, $j=\{1,2,\dots ,m\}$, $m$ is the number of density classes
• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)
• $\alpha _{\rm {S}}(\rho _{j})$ is the sharpness parameter of the classification curve
• $d_{\rm {50;S}}(\rho _{j})$ is the size of a particle of density $\rho _{j}$ which has a 50% probability of reporting to the concentrate stream (mm)
• $R_{\rm {P}}(\rho _{j})$ is the fraction of fine particles of density $\rho _{j}$ which are entrained by the classification action towards the inner section of the spiral trough (frac)
• $\alpha _{\rm {P}}(\rho _{j})$ is the sharpness parameter of the filtration curve
• $d_{\rm {50;P}}(\rho _{j})$ is the size of a particle of density $\rho _{j}$ for which 50% are rejected for classification by the Bagnold (or other counteracting) force (mm)
• $Y_{ij}$ is the fraction of the mass of particles in size class $i$ and density class $j$ which are partitioned (recovered) to the concentrate stream (frac)

A separate set of $\alpha _{\rm {S}}$, $d_{\rm {50;S}}$, $R_{\rm {P}}$, $\alpha _{\rm {P}}$, and $d_{\rm {50;P}}$ parameters are required for each particle with a distinct density (e.g. mineral species).

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. That is,

$y_{ij,s}={\begin{cases}Y_{ij,s}&s=1\\\max \left(0,Y_{ij,s}-Y_{ij,s-1}\right)&s>1\\\end{cases}}$

where:

• $s$ is the index of the product stream, i.e. $s=1$ is the first concentrate stream, $s>1$ are subsequent lower-grade concentrate or middlings streams
• $y_{ij,s}$ is the mass fraction of particles in the feed stream in size class $i$ and density class $j$ which are partitioned to the product stream $s$ (frac)
• $Y_{ij,s}$ is the cumulative mass fraction of particles in the feed stream in size class $i$ and density class $j$ which are partitioned to all the products streams up to and including $s$ (frac)
• The $\max$ 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 Bazin spiral partition model may be invoked from the Excel formula bar with the following function call:

=mdPartition_Bazin(MeanSize as Range, alphaS as Double, d50S as Double, Rp as Double, alphP as Double, d50P as Double)

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:

${\begin{aligned}MeanSize&={\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\\\end{bmatrix}}\\\\alphaS&={\big [}\alpha _{S}(\rho _{j}){\big ]}\\d50S&={\big [}d_{50;{\rm {S}}}(\rho _{j}){\text{ (mm)}}{\big ]}\\Rp&={\big [}R_{\rm {p}}(\rho _{j}){\text{ (frac)}}{\big ]}\\alphaP&={\big [}\alpha _{P}(\rho _{j}){\big ]}\\d50P&={\big [}d_{50;{\rm {P}}}(\rho _{j}){\text{ (mm)}}{\big ]}\\\end{aligned}}$

$mdPartition\_Bazin={\begin{bmatrix}Y_{1j}{\text{ (frac)}}\\\vdots \\Y_{nj}{\text{ (frac)}}\\\end{bmatrix}}$

Figure 2. Example showing the selection of the input parameters (shaded cells), and Results (light blue frame) array in Excel.

SysCAD

The Bazin partition is available from the MetDynamics*GravityConcentrator unit model.

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 a Gravity Concentrator unit 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.
NumParallelUnits	Input	The number of parallel, identical units to simulate: Feed is divided by the number of parallel units before being sent to the unit model. Unit model products are multiplied back by the same value and returned to the SysCAD product streams. All unit model result values are shown per parallel unit.
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 required model method and associated 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 (Rho50-Ep)	The partition to concentrate for each size interval is defined by the Rho50-Ep model.
	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.
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Bazin
OreSpecific	CheckBox	Ore-specific partitions, allows the Bazin equation parameters to be separately input for all species. Default is all species have a separate set of input properties. This option is only available if there is more than one species in the project with the size distribution property.
Parameters
AlphaS	Input	The Bazin equation alphaS parameter for the classification curve.
d50S	Input	The Bazin equation d50S cut size parameter for the classification curve.
Rp	Input	The Bazin equation Rp parameter for the filtration curve.
AlphaP	Input	The Bazin equation alphaP parameter for the filtration curve.
d50P	Input	The Bazin equation d50P cut size parameter for the filtration curve.
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	Actual overall cumulative partition to concentrates of all solid species, for each size interval. Excludes solid species not present in the gravity concentrator feed.
CumulativePartition	Display	Cumulative partition to concentrates for each size interval, in each solid species, as determined by the selected model or user defined value. Note: These values are displayed regardless of whether the solid species is present in the gravity concentrator feed or not.
All (All row, All column)	Display	Displays the actual, total, cumulative partition of all solids with a particle size distribution property in the feed to concentrates. Excludes solid species not present in the gravity concentrator feed.
All (All row, per species)	Display	Actual overall cumulative partition to concentrates for each solid species, for all size intervals in that species. Excludes solid species not present in the gravity concentrator feed.
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	Actual overall partition to concentrate of all solid species, for each size interval. Excludes solid species not present in the gravity concentrator feed.
Partition	Display	Partition to concentrate for each size interval, in each solid species, as determined by the selected model or user defined value. Note: These values are displayed regardless of whether the solid species is present in the gravity concentrator feed or not.
All (All row, All column)	Display	Displays the actual, total, partition of all solids with a particle size distribution property in the feed to concentrate. Excludes solid species not present in the gravity concentrator feed.
All (All row, per species)	Display	Actual overall partition to concentrate for each solid species, for all size intervals in that species. Excludes solid species not present in the gravity concentrator feed.

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 stream without split.