Partition (Density, Stochastic)

Description

This article describes Rao's Stochastic equation for the partition of particles by size and density in gravity separators.

Model theory

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Excel

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The Stochastic partition equation may be invoked from the Excel formula bar with the following function call:

=mdPartition_StochasticRao(MeanSize as Range, RD as Range, A as Double, c as Double, B as Double, rhop as Double, hf as Double, D as double, hl as Double, t 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:

${\begin{aligned}MeanSize&={\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\\\end{bmatrix}}\\\\RD&={\begin{bmatrix}(\rho _{\rm {S}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}}\\\\A&={\big [}A{\big ]}\\c&={\big [}c{\big ]}\\B&={\big [}B{\big ]}\\rhop&={\big [}\rho _{\rm {p}}{\text{ (t/m}}^{3}{\text{))}}{\big ]}\\hf&={\big [}h_{\rm {f}}{\text{ (m/m)}}{\big ]}\\D&={\big [}D{\big ]}\\hl&={\big [}h_{\rm {l}}{\text{ (m/m)}}{\big ]}\\t&={\big [}t{\text{ (s)}}{\big ]}\\returnPartitionMetrics&={\big [}({\text{True / False)}}{\big ]}\\\\\\mdPartition\_StochasticRao&={\begin{bmatrix}{\begin{bmatrix}Y_{11}{\text{ (t/h)}}&\dots &Y_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\Y_{n1}{\text{ (t/h)}}&\dots &Y_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(\rho _{50})_{1}{\text{ (t/m}}^{3}{\text{)}}\\\vdots \\(\rho _{50})_{n}{\text{ (t/m}}^{3}{\text{)}}\\\end{bmatrix}}&{\begin{bmatrix}(E_{\rm {p}})_{1}{\text{ (t/m}}^{3}{\text{)}}\\\vdots \\(E_{\rm {p}})_{n}{\text{ (t/m}}^{3}{\text{)}}\\\end{bmatrix}}&{\begin{array}{c}{\begin{bmatrix}(d_{50})_{1}{\text{ (mm)}}&\dots &(d_{50})_{m}{\text{ (mm)}}\\\end{bmatrix}}^{\rm {*}}\\\\\\\\\end{array}}\end{bmatrix}}\end{aligned}}$ where: $n$ is the number of size intervals $*$ indicates optional results returned if $returnPartitionMetrics=True$

Figure 2. Example showing the selection of the input parameters (purple, green, pink, brown, teal, blue, red and purple cells), and the MeanSize (blue frame), RD (red frame) and Results (light blue frame) arrays in Excel. The parameter $returnPartitionMetrics={\text{True}}$ in this example.

SysCAD

The Stochastic 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.

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.

Stochastic page

The Stochastic page is used to specify the input parameters for the Stochastic model.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Stochastic
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Parameters
A	Input	Partition surface parameter, gravitation and viscous forces.
c	Input	Partition surface parameter, turbulence.
B	Input	Partition surface parameter, fluid drift force.
PivotDensity / Rhop	Input	Density at pivot point.
hf	Input	Feed particle input bed position.
D	Input	Partition surface parameter, standard deviation of particle velocities.
hf	Input	Slice position of the particle bed.
t	Input	Separation period of particles.
Liquids
LiquidsSeparMethod	Split To Con (User)	Liquids are split to concentrate by a user-defined fraction of liquids in the feed.
	Split To Con (Model)	Liquids are split to concentrate by a fraction determined by the selected model.
	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.
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 overflow of all solid species, for each size interval. Excludes solid species not present in the cyclone feed.
Partition	Display	Partition to overflow 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 cyclone 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 overflow. Excludes solid species not present in the cyclone feed.
All (All row, per species)	Display	Actual overall partition to overflow for each solid species, for all size intervals in that species. Excludes solid species not present in the cyclone feed.
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.