View source for Partition (Size, Whiten-Beta)

== Description ==

This article describes Whiten's expression for the efficiency of recovery to overflow, modified to represent observed 'fish hook' behaviour in the finer sizes.{{Napier-Munn et al. (1996)}}

== Model theory ==

[[File:PartitionWhitenBeta1.png|thumb|450px|Figure 1. Whiten-Beta partitions to overflow, with (orange line) and without (blue line) use of the <math>\beta</math> parameter.]]

The '''Whiten-Beta''' expression for partition to overflow is:

{{Model theory (Text, Whiten-Beta Efficiency Curve)}}

The value of <math>\beta^*</math> is not an input parameter and is computed numerically, by iteration, since by rearrangement:

:<math>\beta^* = \dfrac{\ln \big[ \exp (\alpha) + 2 \beta \beta^* (\exp (\alpha) - 1) \big]}{\alpha}</math>

When the value of <math>\beta</math> is zero, the Whiten-Beta equation reverts to:

:<math>E_{{\rm oa}i} = C \left [ \dfrac{ \exp (\alpha) - 1}{\exp \left ( \alpha \dfrac{\bar d_i}{d_{\rm 50c}} \right ) + \exp (\alpha) - 2} \right ]</math>

== Excel ==

The Whiten-Beta efficiency curve model may be invoked from the Excel formula bar with the following function call:

<syntaxhighlight lang="vb">=mdPartition_WhitenBeta(MeanSize as Range, alpha as Double, d50c as Double, C as double, Optional beta as Double = 0, Optional returnBetaStar as Boolean = 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}_n\text{ (mm)}\\
\end{bmatrix}\\
\\
alpha & = \big [\alpha \big ]\\
d50c & = \big [d_{50 \rm c} \text{ (mm)}\big ]\\
C & = \big [C\text{ (frac)}\big ]\\
Beta & = \big [\beta \big ]\\
returnBetaStar & = \big [(\text{True / False)} \big ]\\
\end{align}</math>
|
:<math>
mdPartition\_WhitenBeta = 
\begin{bmatrix}
(E_{\rm oa})_1\text{ (frac)}
&
\beta^*\\
\vdots
&
[-]
\\
(E_{\rm oa})_n\text{ (frac}
&
[-]
\\
\end{bmatrix}
</math>

|rowspan="2"| 
:[[File:PartitionWhitenBeta2.png|frame|Figure 2. Example showing the selection of the input parameters (shaded cells), and '''Results''' (light blue frame) array in Excel. The parameter <math>returnBetaStar=\text{False}</math> in this example.]]
|- style="vertical-align:top;"
|colspan="2"|where <math>\beta^*</math> is omitted from the results array if <math>returnBetaStar=\text{False}</math>.
<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
|}

== SysCAD ==

The Whiten-Beta partition is available from the '''{{Name (Text, Company Name)|nospace=1}}*Hydrocyclone''' unit model.

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

==== Cyclone page ====

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

{{SysCAD (Text, Table Header)}}

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

{{SysCAD (Text, Help Link)}}

|- style="vertical-align:top;"
|OreSpecific
|CheckBox
|
* Ore-specific partitions, allows the Whiten-Beta equation parameters to be separately input for all species.
* Default is all species have the same set of single input properties.
* This option is only available if there is more than one species in the project with the size distribution property.
|-
|Alpha
|Input
|The Whiten-Beta equation alpha parameter.
|-
|d50c
|Input
|The Whiten-Beta equation d50c corrected cut size parameter.
|-
|C
|Input
|The Whiten-Beta equation C parameter.
|-
|Beta
|Input
|The Whiten-Beta equation beta parameter.
|-
|Beta*
|style="background: #eaecf0" | Display
|The Whiten-Beta equation Beta* parameter corresponding to the selected value of Beta.

{{SysCAD (Text, Hydrocyclone, Liquids)|method=1}}

|}

== References ==

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