Jig (King)

Description

This article describes a model for the equilibrium stratification of particles in autogenous media gravity concentration devices such as jigs. The original model formulation is outlined by King (2012) and extensions for particle size effects are proposed by Rao (2007).

Model theory

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

Excel

The King stratification jig model may be invoked from the Excel formula bar with the following function call:

=mdUnit_Jig_King(Parameters as Range, Size as Range, Density As Range, Feed as Range, Optional returnConcProfile As Boolean = False

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:

$Parameters={\begin{bmatrix}{\text{Method}}\\A{\text{ (-)}}\\b{\text{ (-)}}\\\kappa {\text{ (-)}}\\p{\text{ (-)}}\\h_{\rm {s}}{\text{ (m/m)}}\\(Q_{\rm {M,F}})_{\rm {L}}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;Density={\begin{bmatrix}(\rho _{\rm {S}})_{11}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{1m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\vdots &\ddots &\vdots \\(\rho _{\rm {S}})_{n1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{nm}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}}\;\;\;{\text{or}}\;\;\;{\begin{bmatrix}(\rho _{\rm {S}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}},\;\;\;\;\;\;$

$Feed={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {returnConcProfile}}={\big [}{\text{(True / False)}}{\big ]}$

where:

• ${\text{Method}}$ indicates whether the jig is in a batch or continuous configuration, (0 = Batch, 1 = Continuous)
• $A$ is the particle size coefficient (-)
• $b$ is the particle size exponent (-)
• $\kappa$ is the velocity profile shape parameter (-)
• $p$ is the number of bed height increments (-)
• $h_{\rm {s}}$ is the relative bed cut height (m/m)
• $Q_{\rm {M,F}}$ is feed solids mass flow rate by size and ore type (t/h)
• $(Q_{\rm {M,F}})_{\rm {L}}$ is the mass flow feed rate of liquids into the cyclone (t/h)
• ${\mathit {returnConcProfile}}$ indicates whether to return the full volume concentration of solids by density class, size and bed height, (True/False)

Note that ${\mathit {Density}}$ 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:

${\mathit {mdUnit\_Jig\_King}}={\begin{bmatrix}{\begin{bmatrix}{\text{Iterations}}\\{\text{Iteration error}}\\R_{\rm {s}}{\text{ (frac)}}\\R_{\rm {f}}{\text{ (frac)}}\\(Q_{\rm {M,C}})_{\rm {L}}{\text{ (t/h)}}\\(Q_{\rm {M,T}})_{\rm {L}}{\text{ (t/h)}}\\\end{bmatrix}}{\begin{array}{cccccc}{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,C}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,C}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,C}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,C}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,T}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,T}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,T}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,T}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(P_{\rm {C}})_{11}{\text{ (frac)}}&\dots &(P_{\rm {C}})_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\(P_{\rm {C}})_{n1}{\text{ (frac)}}&\dots &(P_{\rm {C}})_{nm}{\text{ (frac)}}\\\end{bmatrix}}&{\begin{bmatrix}(P_{\rm {C}})_{1}{\text{ (frac)}}\\\vdots \\(P_{\rm {C}})_{n}{\text{ (frac)}}\end{bmatrix}}&{\begin{bmatrix}(P_{\rm {C}})_{1}&\dots &(P_{\rm {C}})_{m}\\\end{bmatrix}}&{\begin{bmatrix}{\begin{bmatrix}C_{111}{\text{ (v/v)}}&\cdots &C_{1m1}{\text{ (v/v)}}\\\end{bmatrix}}&\cdots &{\begin{bmatrix}C_{111}{\text{ (v/v)}}&\cdots &C_{nm1}{\text{ (v/v)}}\\\end{bmatrix}}\\\vdots &\ddots &\vdots \\{\begin{bmatrix}C_{11p}{\text{ (v/v)}}&\cdots &C_{1mp}{\text{ (v/v)}}\\\end{bmatrix}}&\cdots &{\begin{bmatrix}C_{11p}{\text{ (v/v)}}&\cdots &C_{nmp}{\text{ (v/v)}}\\\end{bmatrix}}\\\end{bmatrix}}^{*}\\\\\\\\\end{array}}\end{bmatrix}}$

where:

• ${\text{Iterations}}$ is the total number of internal iterations required across all stages to reach the equilibrium stratification condition
• ${\text{Iteration error}}$ is the maximum convergence error of the equilibrium stratification solutions across all stages
• $R_{\rm {s}}$ is the overall recovery of solids to the concentrate (heavy) stream (frac)
• $R_{\rm {f}}$ is the estimated recovery of water to the concentrate stream (frac)
• $(Q_{\rm {M,C}})_{\rm {L}}$ is the mass flow rate of liquids to the concentrate stream (t/h)
• $(Q_{\rm {M,T}})_{\rm {L}}$ is the mass flow rate of liquids to the tailing (light) stream (t/h)
• $Q_{\rm {M,C}}$ is mass flow rate of solids to the concentrate stream (t/h)
• $Q_{\rm {M,T}}$ is mass flow rate of solids to the tailing stream (t/h)
• $P_{\rm {C}}$ is partition fraction of feed solids to the concentrate stream (frac)
• $C_{ijk}$ is the volume concentration of solids of density class $i$ and size interval $j$ at height increment $k$ in the equilibrium-stratified bed (v/v)
• $^{*}$ indicates optional results returned if ${\mathit {returnConcProfile}}={\text{True}}$

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), Density (purple frame) and Feed (green frame) arrays in Excel.
Figure 3. Example showing the outline of the Results (light blue frame) array in Excel. This exampple excludes the concentration profile data (${\mathit {returnConcProfile}}={\text{False}}$)

SysCAD

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 for each connected concentrate discharge stream.

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 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 (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.
Parameters
Method	Batch	The jig is operated as a batch or plug flow device.
	Continuous	The jig is operated as a continuous device.
ParticleSizeCoeff / A	Input	Coefficient of the particle size power law equation.
ParticleSizeExp / b	Input	Exponent of the particle size power law equation.
Kappa	Input	Velocity profile shape parameter.
NumHeightInc / hInc	Input	The number of height increments subdividing the bed.
CutHeight / hs	Input	The relative bed cut height.
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.
Results
Iterations	Display	Shows the number of internal model iterations (per SysCAD step) required to converge the stratification model.
MaxError	Display	Shows the quantity of the convergence error between internal model iterations.

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 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.