WHIMS (Dobby and Finch)

Description

This article describes the Dobby and Finch (1977) model for Wet High Intensity Magnetic Separator (WHIMS) and High Gradient Magnetic Separator (HGMS) machines.

Model theory

Dobby and Finch (1977) describe an empirical model of a laboratory-scale High Gradient Magnetic Separator (HGMS), which may be applicable to Wet High Intensity Magnetic Separators (WHIMS) and other similar machines.

The authors suggest the total recovery of material to the magnetics stream, $R_{\rm {T}}$ (frac), is the sum of recovery by the mechanisms of magnetic capture, $R_{\rm {M}}$ (frac), and physical capture by entrapment, $R_{\rm {P}}$ (frac). For a particle within a given size interval $i$ and magnetic susceptibility class $j$, the total recovery is:

$(R_{\rm {T}})_{ij}=(R_{\rm {M}})_{ij}+(R_{\rm {P}})_{i}$

The magnetic recovery ($R_{\rm {M}}$) is related to particle and separating machine properties by the following relations:

$(R_{\rm {M}})_{ij}=0.5+B\log \left[{\dfrac {(M_{\rm {L}})_{ij}}{M_{50}}}\right]$

where $B$ is the recovery curve slope and $M_{50}$ is the magnetic cut point. Both $B$ and $M_{50}$ are machine-specific parameters, independent of the material being separated.

For free-draining separation chambers, the machine and material parameters are related by the group $M_{\rm {L}}$:

$(M_{\rm {L}})_{ij}={\dfrac {HG(\rho _{ij}\chi _{ij})^{\beta }{{\bar {d}}_{i}}^{1.1}}{u^{1.3}{L_{\rm {m}}}^{1.1}}},\quad \quad G=\min\{H,H_{\rm {s}}\}$

where:

• $H$ is the applied magnetic field strength (T)
• $G$ is the magnetic field gradient (T)
• $H_{\rm {s}}$ is the saturated magnetic field strength (T)
• $\rho$ is the particle density (kg/m³)
• $\chi$ is the mass magnetic susceptibility (m³/kg)
• $\beta$ is the material magnetic susceptibility exponent
• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (m)
• $u$ is the mean interstitial velocity of slurry though the chamber (m/s)
• $L_{\rm {m}}$ is the matrix loading, the ratio of the mass of magnetics in the chamber to the mass of the matrix (kg/kg)

Dobby and Finch found the value of $\beta$ was 1.8 for a group of minerals with a wide susceptibly range, which included haematite, ilmenite, sphalerite and chalcopyrite. When the low susceptibity sphalerite and chalcopyrite were removed from the regression procedure, a better fit was obtained with a $\beta$ of 3.9. The value of $\beta$ may require re-fitting to specific data sets when applying the model.

The recovery by physical capture per size interval, $(R_{\rm {P}})_{i}$ is specified by the user.

Excel

The Dobby and Finch WHIMS model may be invoked from the Excel formula bar with the following function call:

=mdUnit_WHIMS_DobbyFinch(Parameters as Range, Size as Range, Feed as Range, Density As Range, Susceptibility As Range, Optional Rp As Range = 0)

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:

${\mathit {Parameters}}={\begin{bmatrix}H{\text{ (T)}}\\H_{\rm {s}}{\text{ (T)}}\\u{\text{ (m/s)}}\\L_{\rm {m}}{\text{ (kg/kg)}}\\M_{50}{\text{ (-)}}\\B{\text{ (-)}}\\\beta {\text{ (-)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {Size}}={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {Feed}}={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;$

${\mathit {Density}}={\begin{bmatrix}(\rho _{\rm {S}})_{11}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{1m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\vdots &\ddots &\vdots \\(\rho _{\rm {S}})_{n1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{nm}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}}\;\;\;{\text{or}}\;\;\;{\begin{bmatrix}(\rho _{\rm {S}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {Susceptibility}}={\begin{bmatrix}\chi _{11}{\text{ (m}}^{\text{3}}{\text{/kg)}}&\dots &\chi _{1m}{\text{ (m}}^{\text{3}}{\text{/kg)}}\\\vdots &\ddots &\vdots \\\chi _{n1}{\text{ (m}}^{\text{3}}{\text{/kg)}}&\dots &\chi _{nm}{\text{ (m}}^{\text{3}}{\text{/kg)}}\\\end{bmatrix}}\;\;\;{\text{or}}\;\;\;{\begin{bmatrix}\chi _{1}{\text{ (m}}^{\text{3}}{\text{/kg)}}&\dots &\chi _{m}{\text{ (m}}^{\text{3}}{\text{/kg)}}\\\end{bmatrix}},\;\;\;\;\;\;$

${\mathit {Rp}}={\begin{bmatrix}(R_{\rm {p}})_{1}\\\vdots \\(R_{\rm {p}})_{n}\\\end{bmatrix}}^{*}$

where:

• $Q_{\rm {M,F}}$ is feed solids mass flow rate by size and ore type (t/h)
• $^{*}$ indicates an optional input, default values of $(R_{\rm {P}})_{i}=0$ for $i=\{1,2,\dots ,n\}$ are assumed if ${\mathit {Rp}}$ is omitted

Note that both ${\mathit {Density}}$ and ${\mathit {Susceptibility}}$ 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:

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

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), Density (green frame), Susceptibility (pink frame) and Rp (brown 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.
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.
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.

WHIMS page

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

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
WHIMS
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Parameters
MagneticFieldStrength / H	Input	Magnetic field strength.
SatMagFieldStrength / Hs	Input	Saturation magnetic field strength.
SlurryVelocity / u	Input	Average velocity of slurry flow through the matrix.
MatrixLoading / Lm	Input	Ratio of the mass of magnetic solids to the mass of the matrix in the separation chamber.
MagneticCutPoint / M50	Input	Magnetic cut point.
RecoveryCurveSlope / B	Input	Slope of the recovery curve, i.e. coefficient of the log term.
SusceptibilityExponent / Beta	Input	Exponent of the susceptibility equation term.
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.

Susceptibility page

This page is used to specify the magnetic mass susceptibility 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.
Susceptibility
Selection	Stream	A single mass susceptibility value is specified for all species and sizes.
	Species	Mass susceptibility values are specified per species, for all sizes of that species.
	Size	Mass susceptibility values are specified per size interval, for all species in that interval.
	Species-Size	Individual mass susceptibility 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.
Rp	Input	Fraction recovered to magnetics by physical capture of particles.
Susceptibility	Input / Display	Mass susceptibility 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.