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== Description ==
== Description ==


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


== Model theory ==
== Model theory ==
Line 9: Line 9:
== Excel ==
== Excel ==


{{Under construction|section}}
The King stratification jig model may be invoked from the Excel formula bar with the following function call:
 
<syntaxhighlight lang="vb">=mdUnit_Jig_King(Parameters as Range, Size as Range, Density As Range, Feed as Range, Optional returnConcProfile As Boolean = False)</syntaxhighlight>
 
{{Excel (Text, Help, No Arguments)}}
 
=== Inputs ===
 
{{Excel (Text, Inputs)}}
 
:<math>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},\;\;\;\;\;\;
</math>
 
 
:<math>
 
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 ]
 
</math>
 
 
where:
* <math>\text{Method}</math> indicates whether the jig is in a batch or continuous configuration, ''(0 = Batch, 1 = Continuous)''
* <math>A</math> is the particle size coefficient (-)
* <math>b</math> is the particle size exponent (-)
* <math>\kappa</math> is the velocity profile shape parameter (-)
* <math>p</math> is the number of bed height increments (-)
* <math>h_{\rm s}</math> is the relative bed cut height (m/m)
* <math>Q_{\rm M,F}</math> is feed solids mass flow rate by size and ore type (t/h)
* <math>(Q_{\rm M,F})_{\rm L}</math> is the mass flow feed rate of liquids into the jig (t/h)
* <math>\mathit{returnConcProfile}</math> indicates whether to return the full volume concentration of solids by density class, size and bed height, ''(True/False'')
 
Note that <math>\mathit{Density}</math> 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:
 
:<math>
\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}
</math>
 
 
where:
* <math>\text{Iterations}</math> is the total number of internal iterations required across all stages to reach the equilibrium stratification condition
* <math>\text{Iteration error}</math> is the maximum convergence error of the equilibrium stratification solutions across all stages
* <math>R_{\rm s}</math> is the overall recovery of solids to the concentrate (heavy) stream (frac)
* <math>R_{\rm f}</math> is the estimated recovery of water to the concentrate stream (frac)
* <math>(Q_{\rm M,C})_{\rm L}</math> is the mass flow rate of liquids to the concentrate stream (t/h)
* <math>(Q_{\rm M,T})_{\rm L}</math> is the mass flow rate of liquids to the tailing (light) stream (t/h)
* <math>Q_{\rm M,C}</math> is mass flow rate of solids to the concentrate stream (t/h)
* <math>Q_{\rm M,T}</math> is mass flow rate of solids to the tailing stream (t/h)
* <math>P_{\rm C}</math> is partition fraction of feed solids to the concentrate stream (frac)
* <math>C_{ijk}</math> is the volume concentration of solids of density class <math>i</math> and size interval <math>j</math> at height increment <math>k</math> in the equilibrium-stratified bed (v/v)
* <math>^*</math> indicates optional results returned if <math>\mathit{returnConcProfile} = \text{True}</math>
 
=== Example ===
 
The images below show the selection of input arrays and output results in the Excel interface.
 
{|
|- style="vertical-align:top;"
| [[File:JigKing1.png|left|frame|Figure 1. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]] || [[File:JigKing2.png|left|frame|Figure 2. Example showing the selection of the '''Size''' (red frame), '''Density''' (purple frame) and '''Feed''' (green frame) arrays in Excel.]] || 
|- style="vertical-align:top;"
|colspan="2"| [[File:JigKing3.png|left|frame|Figure 3. Example showing the outline of the '''Results''' (light blue frame) array in Excel. This exampple excludes the concentration profile data (<math>\mathit{returnConcProfile} = \text{False}</math>)]]
|}


== SysCAD ==
== SysCAD ==


{{Under construction|section}}
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.
 
{{SysCAD (Page, Gravity Concentrator, DLL*GravityConcentrator)}}
 
==== Con page ====
 
The Con page is used to specify the required model method and associated input parameters.
 
{{SysCAD (Text, Table Header)}}
 
{{SysCAD (Text, Gravity Concentrator, Con)}}
 
{{SysCAD (Text, Help Link)}}
 
|-
! colspan="3" style="text-align:left;" |''Parameters''
|- style="vertical-align:top;"
|rowspan="2" | Method
|Batch
|The jig is operated as a batch or plug flow device.
|- style="vertical-align:top;"
|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.
 
{{SysCAD (Text, Gravity Concentrator, Liquids)|method=0}}
 
|-
! colspan="3" style="text-align:left;" |''Results''
|-
|Iterations
|style="background: #eaecf0" |Display
|Shows the number of internal model iterations (per SysCAD step) required to converge the stratification model.
|-
|MaxError
|style="background: #eaecf0" |Display
|Shows the quantity of the convergence error between internal model iterations.
 
|}
 
{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Partition|ActionL=partition|DestinationU=Concentrate|DestinationL=concentrate|UnitL=gravity concentrator|GravityMetrics=false}}
 
{{SysCAD (Page, About)}}
 
==== Additional notes ====
 
{{SysCAD (Text, No PSD Splits)|gasstream=tail}}


== References ==
== References ==

Latest revision as of 10:47, 14 September 2024

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).[1][2]

Model theory

Under construction icon-blue.svg.png 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 () x column () format:



where:

  • indicates whether the jig is in a batch or continuous configuration, (0 = Batch, 1 = Continuous)
  • is the particle size coefficient (-)
  • is the particle size exponent (-)
  • is the velocity profile shape parameter (-)
  • is the number of bed height increments (-)
  • is the relative bed cut height (m/m)
  • is feed solids mass flow rate by size and ore type (t/h)
  • is the mass flow feed rate of liquids into the jig (t/h)
  • indicates whether to return the full volume concentration of solids by density class, size and bed height, (True/False)

Note that 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:


where:

  • is the total number of internal iterations required across all stages to reach the equilibrium stratification condition
  • is the maximum convergence error of the equilibrium stratification solutions across all stages
  • is the overall recovery of solids to the concentrate (heavy) stream (frac)
  • is the estimated recovery of water to the concentrate stream (frac)
  • is the mass flow rate of liquids to the concentrate stream (t/h)
  • is the mass flow rate of liquids to the tailing (light) stream (t/h)
  • is mass flow rate of solids to the concentrate stream (t/h)
  • is mass flow rate of solids to the tailing stream (t/h)
  • is partition fraction of feed solids to the concentrate stream (frac)
  • is the volume concentration of solids of density class and size interval at height increment in the equilibrium-stratified bed (v/v)
  • indicates optional results returned if

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 ()

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 ButtonModelHelp.png 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 ButtonLicensingHelp.png Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information ButtonCopyToClipboard.png 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 ButtonBrowse.png 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.

References

  1. ↑ King, R.P., 2012. Modeling and Simulation of Mineral Processing Systems. Elsevier.
  2. ↑ Rao, B.V., 2007. Extension of particle stratification model to incorporate particle size effects. International Journal of Mineral Processing, 85(1-3), pp.50-58.
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