View source for 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).{{King (2012)}}{{Rao (2007)}}

== Model theory ==

{{Under construction|section}}

== Excel ==

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 ==

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 ==

[[Category:Excel]]
[[Category:SysCAD]]