View source for Partition (Density, Pivot)

== Description ==

This article describes the '''Pivot model''' equation for the partition of particles by size and density in a dense medium cyclone.{{Scott and Napier-Munn (1992)}}

The Pivot model may also be applicable to other dense medium and gravity concentration processes, based indirectly on Rao et al.'s (2003) derivation of the Pivot model from the [[Partition (Density, Stochastic)|Stochastic model]] and the presented data analyses.{{Rao et al. (2003)}}

== Model theory ==

[[File:PartitionPivot2.png|thumb|500px|Figure 1. Pivot model partition to sinks. Each coloured line represents a particle class with the geometric mean size indicated in the series label. The pivot point is evident at a density of 2.0 t/m<sup>3</sup>.]]

[[File:PartitionPivot3.png|thumb|500px|Figure 2. Pivot model partition to sinks, using the same model parameters as Figure 1. Each coloured line represents a density class with the specific gravity indicated in the series label.]]

Scott and Napier-Munn (1992) derived an empirical equation that describes the partition of particles of a given size and density to the sinks stream of a dense medium cyclone.{{Scott and Napier-Munn (1992)}} 

Scott and Napier-Munn's approach is based on the observed ''pivot phenomenon'' where the partition curves of particles of all sizes appear to 'pivot' at a unique density value, as illustrated in Figure 1.

The proposed ''Pivot model'' equation is:

:<math>Y_{ij} = \dfrac{1}{1 + \exp \left \{ \ln ({Y_{\rm p}}^{-1} - 1) + \ln 3 \left (\dfrac{\rho_{\rm p} - (\rho_{\rm S})_{ij}}{k \bar {d_i}^n} \right )\right \}}</math>

where:
* <math>i</math> is the index of the size interval, <math>i = \{1,2,\dots,q\}</math>, <math>q</math> is the number of size intervals
* <math>j</math> is the index of the density class, <math>j = \{1,2,\dots,m\}</math>, <math>m</math> is the number of density classes
* <math>Y_{ij}</math> is the fraction of the mass of particles in size class <math>i</math> and density class <math>j</math> which are partitioned (recovered) to the sinks stream (frac)
* <math>\rho_{\rm p}</math> is the pivot density (t/m<sup>3</sup>)
* <math>\rho_{\rm S}</math> is the particle density (t/m<sup>3</sup>)
* <math>Y_{\rm p}</math> is the partition to sinks at the pivot density (frac)
* <math>k</math> is the Ecart Probable size coefficient
* <math>n</math> is the Ecart Probable size exponent
* <math>\bar d_{i}</math> is the [[Conversions|geometric mean size]] of particles in size interval <math>i</math> (mm)

===Partition metrics ===

Several metrics are provided to characterise the Pivot model partition curve.

The <math>d_{50}</math>, also known as the ''cut or separation size'', is defined as the size of a particle of a given density which has an even (50%) chance of appearing in either the sinks or floats stream. The <math>d _{50}</math> size is obtained by rearranging the Pivot model:

:<math>d_{50} = \left ( \dfrac{\ln 3}{k} \dfrac{(\rho_j - \rho_{\rm p})}{\ln ({Y_{\rm p}}^{-1} - 1)} \right ) ^ {\frac{1}{n}}</math>

which is only defined when <math>(\rho_j > \rho_{\rm p}, Y_{\rm p} \leq 0.5)</math> or <math>(\rho_j < \rho_{\rm p}, Y_{\rm p} \geq 0.5)</math>

The <math>\rho_{50}</math>, also known as the ''cut or separation density'', is defined as the density of a particle of a given size which has an even (50%) chance of appearing in either the sinks or floats stream. The <math>\rho _{50}</math> density is obtained by rearranging the Pivot model:

:<math>\rho_{50} = \rho_{\rm p} + \dfrac{k \bar {d_i}^n}{\ln 3} \ln ( {Y_{\rm p}}^{-1} - 1)</math>

The ''Ecart Probable'', or <math>E_{\rm p}</math>, is a measure of the deviation of a partition curve from a perfect separation, and is typically defined for density separations as:

:<math>E_{\rm p} = \dfrac{\rho_{75} - \rho_{25}}{2}</math>

where <math>\rho_{75}</math> and <math>\rho_{25}</math> are the densities of particles which have a 75% and 25% probability, respectively, of appearing in the sinks stream.

{{Model theory (Text, Gravity Concentrator, Middlings)}}

== Excel ==

The Pivot partition equation may be invoked from the Excel formula bar with the following function call:

<syntaxhighlight lang="vb">=mdPartition_Pivot(MeanSize as Range, RD as Range, Yp as Double, rhop as Double, k as Double, n as Double, Optional returnPartitionMetrics as Bool = false)</syntaxhighlight>

{{Excel (Text, Help, No Arguments)}}

The input parameters and model results are defined below in matrix notation, along with an example image showing the selection of the same cells in the Excel interface:

:{| 
|- style="vertical-align:top;"
|
<math>
\begin{align}
MeanSize & = 
\begin{bmatrix}
\bar{d}_1\text{ (mm)}\\
\vdots\\
\bar{d}_q\text{ (mm)}\\
\end{bmatrix}\\
\\
RD & = \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}\\
\\
Y_{\rm p} & = \big [Y_{\rm p} \text{ (frac)} \big ]\\
rhop & = \big [\rho_{\rm p} \text{ (t/m}^3\text{))}\big ]\\
k & = \big [k\text{ (-)} \big ]\\
n & = \big [n\text{ (-)} \big ]\\
returnPartitionMetrics & = \big [(\text{True / False)} \big ]\\
\\
\\
mdPartition\_Pivot & = 
\begin{bmatrix}
\begin{bmatrix}
Y_{11}\text{ (t/h)} & \dots & Y_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
Y_{q1}\text{ (t/h)} & \dots & Y_{qm}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{bmatrix}
(\rho_{50})_1\text{ (t/m}^3\text{)}\\
\vdots\\
(\rho_{50})_q\text{ (t/m}^3\text{)}\\
\end{bmatrix}^{\rm *}

&

\begin{bmatrix}
(E_{\rm p})_1\text{ (t/m}^3\text{)}\\
\vdots\\
(E_{\rm p})_q\text{ (t/m}^3\text{)}\\
\end{bmatrix}^{\rm *}

&

\begin{array}{c}
\begin{bmatrix}
(d_{50})_{1}\text{ (mm)} & \dots & (d_{50})_m\text{ (mm)}\\ 
\end{bmatrix}^{\rm *}
\\
\\
\\
\\
\end{array}

\end{bmatrix}
\end{align}</math>


where:
* <math>n</math> is the number of size intervals
* <math>m</math> is the number of density classes
* <math>^*</math> indicates optional results returned if <math>\mathit{returnPartitionMetrics} = \text{True}</math> 


|- style="vertical-align:top;"
| [[File:PartitionPivot1.png|frame|Figure 3. Example showing the selection of the input parameters (purple, green, pink and brown cells), and the '''MeanSize''' (blue frame), '''RD''' (red frame) and '''Results''' (light blue frame) arrays in Excel. The parameter <math>\mathit{returnPartitionMetrics} =\text{True}</math> in this example.]]
|}

== SysCAD ==

The Pivot partition is available from the '''{{Name (Text, Company Name)|nospace=1}}*GravityConcentrator''' and '''{{Name (Text, Company Name)|nospace=1}}*DenseMedium''' unit models.

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 provided for each connected concentrate discharge stream of the '''{{Name (Text, Company Name)|nospace=1}}*GravityConcentrator''' model.

{{SysCAD (Page, Gravity Concentrator, DLL*GravityConcentrator)}}

==== Con page ====

The Con page is used to specify the selected model input parameters.

{{SysCAD (Text, Table Header)}}

{{SysCAD (Text, Gravity Concentrator, Con)}}

{{SysCAD (Text, Help Link)}}

{{SysCAD (Text, Partition, Pivot)}}

{{SysCAD (Text, Gravity Concentrator, Liquids)|method=0}}

|}

{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Partition|ActionL=partition|DestinationU=Concentrate|DestinationL=concentrate|UnitL=gravity concentrator|GravityMetrics=true|Cumulative=true}}

{{SysCAD (Page, Dense Medium, DLL*DenseMedium)}}

==== Pivot page ====

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

{{SysCAD (Text, Table Header)}}

|-
! colspan="3" style="text-align:left;" |''Pivot''

{{SysCAD (Text, Help Link)}}

{{SysCAD (Text, Partition, Pivot)}}

{{SysCAD (Text, Dense Medium, Liquids)|method=0}}

|}

{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Partition|ActionL=partition|DestinationU=Sinks|DestinationL=sinks|UnitL=dense medium separator|GravityMetrics=true}}

{{SysCAD (Page, About)}}

==== Additional notes ====

{{SysCAD (Text, No PSD Splits)|gasstream=tail/floats}}

== See also ==
* [[Partition (Density, Stochastic)]]

== References ==

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