View source for Partition (Size, Bazin)

== Description ==

This article describes the Bazin et al. (2014) empirical expression for the recovery of particles to the concentrate stream of a spiral concentrator.{{Bazin et al. (2014)}}

The form of the empirical equation may also be applicable to the Falcon gravity concentrator in some circumstances, as observed by Dehaine et al (2016).{{Dehaine et al (2016)}}

== Model theory ==

[[File:PartitionBazin1.png|thumb|450px|Figure 1. Bazin partitions to concentrate, for three minerals with differing densities.]]

Bazin et al. (2014) proposed an empirical expression for the partition of particles of a given size and density to the concentrate stream of a spiral concentrator. The concentrate stream is defined here as the inner part of the spiral trough.

The expression is formulated as the product of two counter-acting mechanisms that simultaneously force heavy minerals to the centre of the trough and coarse particles to the outer edge of the trough.

The partition of particles of a given size and density to the concentrate stream, <math>y(d, \rho)</math> (frac), is:

:<math>y(d, \rho) = S_{\rm c}(d, \rho) \times P_{\rm R}(d, \rho)</math>

where:
* <math>S_{\rm c}(d, \rho)</math> is the fraction of particles which should nominally report to the inner part of the trough (concentrate stream) due to gravitational forces, the ''classification'' action
* <math>P_{\rm R}(d, \rho)</math> is the probability that a particle resists the inward forces and is directed towards the outer trough by the Bagnold force, the ''filtration'' action
* <math>d</math> and <math>\rho</math> are the diameter (mm) and density (t/m<sup>3</sup>) of the particle respectively

For series of size intervals, the <math>S_{\rm c}(d, \rho)</math> and <math>P_{\rm R}(d, \rho)</math> terms are computed as:

:<math>S_{\rm c}(d_i, \rho_j) = \dfrac{\exp \left( \alpha_{\rm S}(\rho_j) \dfrac{\bar d_i}{d_{50;\rm S}(\rho_j)}\right ) - 1}{\exp \left( \alpha_{\rm S}(\rho_j) \dfrac{\bar d_i}{d_{50;\rm S}(\rho_j)}\right ) + \exp \big( \alpha_{\rm S}(\rho_j)\big) - 2}</math>

:<math>P_{\rm R}(d_i, \rho_j) = R_{\rm P}(\rho_j) + (1 - R_{\rm P}(\rho_j))\left \{ 1 - \left [ \dfrac{\exp \left( \alpha_{\rm P}(\rho_j) \dfrac{\bar d_i}{d_{50;\rm P}(\rho_j)}\right ) - 1}{\exp \left( \alpha_{\rm P}(\rho_j) \dfrac{\bar d_i}{d_{50;\rm P}(\rho_j)}\right ) + \exp \big( \alpha_{\rm P}(\rho_j)\big) - 2} \right ] \right \}</math>

where:
* <math>i</math> is the index of the size interval, <math>i = \{1,2,\dots,n\}</math>, <math>n</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>\bar d_{i}</math> is the [[Conversions|geometric mean size]] of particles in size interval <math>i</math> (mm)
* <math>\alpha_{\rm S}(\rho)</math> is the sharpness parameter of the classification curve
* <math>d_{\rm 50;S}(\rho)</math> is the size of a particle of density <math>d\rho</math> which has a 50% probability of reporting to the concentrate stream (mm)
* <math>R_{\rm P}</math> is the fraction of fine particles which are entrained by the classification action towards the inner section of the spiral trough (frac)
* <math>\alpha_{\rm P}(\rho)</math> is the sharpness parameter of the filtration curve
* <math>d_{\rm 50;P}(\rho)</math> is the size of a particle of density <math>d\rho</math> for which 50% are rejected for classification by the Bagnold force (mm)

A separate set of <math>\alpha_{\rm S}</math>, <math>d_{\rm 50;S}</math>, <math>R_{\rm P}</math>, <math>\alpha_{\rm P}</math>, and <math>d_{\rm 50;P}</math> parameters are required for each particle with a distinct density (e.g. mineral species).

== Excel ==

The Bazin spiral partition model may be invoked from the Excel formula bar with the following function call:

<syntaxhighlight lang="vb">=mdPartition_Bazin(MeanSize as Range, alphaS as Double, d50S as Double, Rp as Double, alphP as Double, d50P as Double)</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}_n\text{ (mm)}\\
\end{bmatrix}\\
\\
alphaS & = \big [\alpha_S(\rho) \big ]\\
d50S & = \big [d_{50; \rm S}(\rho) \text{ (mm)}\big ]\\
Rp & = \big [R_{\rm p}(\rho)\text{ (frac)}\big ]\\
alphaP & = \big [\alpha_P(\rho) \big ]\\
d50P & = \big [d_{50; \rm P}(\rho) \text{ (mm)}\big ]\\
\end{align}</math>
|
:<math>
mdPartition\_Bazin = 
\begin{bmatrix}
(y(d_1, \rho)\text{ (frac)}
\\
\vdots
\\
(y(d_n, \rho)\text{ (frac}
\\
\end{bmatrix}
</math>

|rowspan="2"| 
:[[File:PartitionBazin2.png|frame|Figure 2. Example showing the selection of the input parameters (shaded cells), and '''Results''' (light blue frame) array in Excel.]]
|}

== SysCAD ==

The Bazin partition is available from the '''{{Name (Text, Company Name)|nospace=1}}*GravityConcentrator''' unit model.

The sections and variable names used in the SysCAD interface are described in detail in the following tables.

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

==== Bazin page ====

The Bazin page is used to specify the input parameters for the Bazin model.

{{SysCAD (Text, Table Header)}}

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

{{SysCAD (Text, Help Link)}}

|- style="vertical-align:top;"
|OreSpecific
|CheckBox
|
* Ore-specific partitions, allows the Bazin equation parameters to be separately input for all species.
* Default is all species have a separate set of input properties.
* This option is only available if there is more than one species in the project with the size distribution property.
|-
! colspan="3" style="text-align:left;" |''Parameters''
|-
|AlphaS
|Input
|The Bazin equation alphaS parameter for the classification curve.
|-
|d50S
|Input
|The Bazin equation d50S cut size parameter for the classification curve.
|-
|Rp
|Input
|The Bazin equation Rp parameter for the filtration curve.
|-
|AlphaP
|Input
|The Bazin equation alphaP parameter for the filtration curve.
|-
|d50P
|Input
|The Bazin equation d50P cut size parameter for the filtration curve.

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

|}

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

{{SysCAD (Page, About)}}

== References ==

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