WLIMS (King and Schneider)

Description

This article describes the King and Schneider (1995) model for drum-type Wet Low Intensity Magnetic Separators (WLIMS).

Model theory

Recovery to non-magnetics

King and Schneider (1995) proposed the following empirical model for recovery to the non-magnetics stream, $R_{\rm {NM}}$ (frac),:

$(R_{\rm {NM}})_{ij}=\alpha _{i}+(1-\alpha _{i})\cdot (E_{\rm {NM}})_{j}$

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 magnetics volumetric grade class, $j=\{1,2,\dots ,m\}$, $m$ is the number of volumetric grade classes
• $\alpha _{i}$ is the fraction of particle mass in size interval $i$ which bypasses directly to the non-magnetics stream (frac)
• $(E_{\rm {NM}})_{j}$ is the fraction of mass in magnetics volumetric grade class $j$ which is recovered to the non-magnetics stream (frac)

The bypass function, $\alpha _{i}$, represents particles which are entrained via water drag to the non-magnetics stream, and is modelled by:

$\alpha _{i}=\kappa \cdot \exp({-\zeta {\bar {d}}_{i}})$

where:

• $\kappa$ is the bypass fraction coefficient (frac)
• $\zeta$ is the bypass exponential factor (mm^-1)
• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)

The primary classification function, $(E_{\rm {NM}})_{j}$, is:

$(E_{\rm {NM}})_{j}=1-\exp \left[{-\ln 2\left({\dfrac {1-(g_{\rm {v}})_{j}}{1-g_{50}}}\right)^{\lambda }}\right]$

where:

• $g_{\rm {v}}$ is the volumetric grade (or fraction) of the magnetics phase in particles belonging to class $j$ (v/v)
• $g_{50}$ is the separation volumetric grade, the grade at which a particle has an equal probability of reporting to either the magnetics or non-magnetics stream (v/v)
• $\lambda$ is the Rosin-Rammler exponent

The Rosin-Rammler exponent, $\lambda$, is related to the Sharpness Index, ${\mathit {SI}}$, by:

${\mathit {SI}}=\exp \left({-{\dfrac {1.5725}{\lambda }}}\right)\implies \lambda =-{\dfrac {1.5725}{\ln {\mathit {SI}}}}$

And finally, recovery to the magnetics stream, $R_{\rm {M}}$ (frac), is:

$(R_{\rm {M}})_{ij}=1-(R_{\rm {NM}})_{ij}$

Recovery to magnetics

This section is currently under construction. Please check back later for updates and revisions.

King and Schneider's empirical WLIMS model appears in the MODSIM simulation software package which is distributed with Modeling and Simulation of Mineral Processing Systems (King, 2012).

Evaluation of the MODSIM version of the WLIMS model suggests a different implementation of the recovery equations. Rather than recovery to non-magnetics, the MODSIM WLIMS model appears to calculate the recovery to magnetics with the following modifications of King and Schneider's original equations:

$(R_{\rm {M}})_{ij}=\alpha _{i}+(1-\alpha _{i})\cdot (E_{\rm {M}})_{j}$

$(E_{\rm {M}})_{j}=1-\exp \left[{-\ln 2\left({\dfrac {(g_{\rm {v}})_{j}}{g_{50}}}\right)^{\lambda }}\right]$

where $E_{\rm {M}}$ is classification to the magnetics stream and all other terms are defined as previously.

In this context, the bypass function, $\alpha$, might be better described as physical capture by particle entrapment to magnetics, rather than entrainment to non-magnetics.

This implementation of King and Schneider's WLIMS model allows a user to select either the original recovery to non-magnetics, or alternative recovery to magnetics equations for maximum flexibility.

Excel

The King and Schneider WLIMS model may be invoked from the Excel formula bar with the following function call:

=mdUnit_WLIMS_KingSchneider(Parameters as Range, Size as Range, Feed as Range, MagneticsVolGrade As Range)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

Inputs

The required inputs are defined below in matrix notation with elements corresponding to cells in Excel row ($i$) x column ($j$) format:

${\mathit {Parameters}}={\begin{bmatrix}{\text{Recover to NM (true/false)}}\\{\mathit {SI}}{\text{ (-)}}\\g_{50}{\text{ (v/v)}}\\\kappa {\text{ (frac)}}\\\zeta {\text{ (mm}}^{-1}{\text{)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {Size}}={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {Feed}}={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;$

${\mathit {MagneticsVolGrade}}={\begin{bmatrix}(g_{\rm {v}})_{11}{\text{ (v/v)}}&\dots &(g_{\rm {v}})_{1m}{\text{ (v/v)}}\\\vdots &\ddots &\vdots \\(g_{\rm {v}})_{n1}{\text{ (v/v)}}&\dots &(g_{\rm {v}})_{nm}{\text{ (v/v)}}\\\end{bmatrix}}\;\;\;{\text{or}}\;\;\;{\begin{bmatrix}(g_{\rm {v}})_{1}{\text{ (v/v)}}&\dots &(g_{\rm {v}})_{m}{\text{ (v/v)}}\\\end{bmatrix}},\;\;\;\;\;\;$

where ${\text{Recover to NM}}$ indicates whether the recovery equations are for King and Schnieder's (1995) original recovery to the non-magnetics stream (true), or the alternative recovery to magnetics stream (false).

Note that ${\mathit {MagneticsVolGrade}}$ can be specified on a per particle size per ore type basis, or with a singular value for all particle sizes of a given ore type.

Results

The results are displayed in Excel as an array corresponding to the matrix notation below:

${\mathit {mdUnit\_WLIMS\_KingSchneider}}={\begin{bmatrix}{\begin{array}{c}{\begin{bmatrix}R_{\rm {M}}{\text{ (frac)}}\\\end{bmatrix}}\\\\\\\end{array}}{\begin{array}{cccccc}{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,M}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,M}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,M}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,M}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,NM}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,NM}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,NM}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,NM}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(R_{\rm {M}})_{11}{\text{ (frac)}}&\dots &(R_{\rm {M}})_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\(R_{\rm {M}})_{n1}{\text{ (frac)}}&\dots &(R_{\rm {M}})_{nm}{\text{ (frac)}}\\\end{bmatrix}}&{\begin{bmatrix}(R_{\rm {M}})_{1}{\text{ (frac)}}\\\vdots \\(R_{\rm {M}})_{n}{\text{ (frac)}}\end{bmatrix}}&{\begin{array}{c}{\begin{bmatrix}(R_{\rm {M}})_{1}&\dots &(R_{\rm {T}})_{m}\\\end{bmatrix}}\\\\\\\end{array}}&{\begin{bmatrix}\alpha _{1}{\text{ (frac)}}\\\vdots \\\alpha _{n}{\text{ (frac)}}\end{bmatrix}}\end{array}}\end{bmatrix}}$

where:

• $Q_{\rm {M,M}}$ is mass flow rate of solids to the magnetics stream (t/h)
• $Q_{\rm {M,NM}}$ is mass flow rate of solids to the non-magnetics stream (t/h)

Example

The images below show the selection of input arrays and output results in the Excel interface.

Figure 1. Example showing the selection of the Parameters (blue frame) array in Excel.	Figure 2. Example showing the selection of the Size (red frame), Feed (purple frame), and VolumetricGrade (green frame) arrays in Excel.
Figure 3. Example showing the outline of the Results (light blue frame) array in Excel.

SysCAD

MD_MagneticSeparator 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 MassFracToMags option appears below.
MassFracToMags	Input	Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the magnetics stream when the model is off.
Method	User Defined	The recovery to magnetics for each size interval is defined by the user.
	WLIMS (King & Schneider)	The recovery to magnetics for each size interval is defined by the King and Schneider model.
	WHIMS (Dobby and Finch)	The recovery to magnetics for each size interval is defined by the Dobby and Finch model.
Options
ShowQFeed	CheckBox	QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
ShowQMags	CheckBox	QMags and associated tab pages (eg Sp) will become visible, showing the properties of the magnetics stream.
ShowQNonMags	CheckBox	QNonMags and associated tab pages (eg Sp) will become visible, showing the properties of the non-magnetics 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.

WLIMS page

The WLIMS page is used to specify the required model method and associated input parameters.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
WLIMS
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Parameters
RecoverToNonMags	Checkbox	Assigns recovery predicted by model equations to non-magnetics or magnetics product stream.
SharpnessIndex / SI	Input	Sharpness Index.
SeparationVolGrade / g50	Input	Separation volume grade, i.e. grade cut point of separation.
BypassFraction / Kappa	Input	Fractional coefficient of the bypass equation.
BypassExpFactor / Zeta	Input	Exponential factor of the bypass equation.
Liquids
LiquidsSeparMethod	Split To Mags (User)	Liquids are split to magnetics by a user-defined fraction of liquids in the feed.
	Mags Solids Fraction	Sufficient liquids mass is recovered to the magnetics stream to yield the user-defined magnetics solids mass fraction value (if possible).
	Mags Liquids Fraction	Sufficient liquids mass is recovered to the magnetics stream to yield the user-defined magnetics liquids mass fraction value (if possible).
MagsSolidsFracReqd / Mags.SfReqd	Input	Required value of the mass fraction of solids in the magnetics stream. Only visible if Mags Solids Fraction is selected.
MagsLiquidsFracReqd / Mags.LfReqd	Input	Required value of the mass fraction of liquids in the magnetics stream. Only visible if Mags Liquids Fraction is selected.
LiqSplitToMags	Input/Display	The fraction of feed liquids recovered to the magnetics stream.

Magnetics page

This page is used to specify the volumetric grade of magnetics phase input parameters.

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.
MagneticsVolGrade
Selection	Stream	A single magnetics volumetric grade value is specified for all species and sizes.
	Species	Magnetics volumetric grade values are specified per species, for all sizes of that species.
	Size	Magnetics volumetric grade values are specified per size interval, for all species in that interval.
	Species-Size	Individual magnetics volumetric grade values are specified for each size interval of each species.
Size	Display	Size of each interval in internal mesh series.
MeanSize	Display	Geometric mean size of each interval in internal mesh series.
MagneticsVolGrade	Input / Display	Magnetics volumetric grade for each size interval, in each solid species.

Recovery page

The Recovery page is used to specify or display the recovery 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.
Recovery
Method	Model/User	Select model-calculated or user-defined recovery 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 recovery to magnetics of all solid species, for each size interval. Excludes solid species not present in the magnetic separator feed.
Recovery	Display	Recovery to magnetics 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 magnetic separator feed or not.
All (All row, All column)	Display	Displays the actual, total, recovery of all solids with a particle size distribution property in the feed to magnetics. Excludes solid species not present in the magnetic separator feed.
All (All row, per species)	Display	Actual overall recovery to magnetics for each solid species, for all size intervals in that species. Excludes solid species not present in the magnetic separator 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 non-magnetics stream without split.