View source for Vibrating Screen (Whiten)

== Description ==

This article describes an implementation of the '''Whiten and White''' (1977) model for multi-row vibrating screen classification, with extensions by Dehghani et al. (2002) and Firth and Hart (2008).{{Whiten and White (1977)}}{{Dehghani et al. (2002)}}{{Firth and Hart (2008)}}

== Model theory ==

[[File:VibratingScreenWhiten1.png|thumb|550px|Figure 1. Example panel layout for a vibrating screen deck.]]

[[File:VibratingScreenWhiten2.png|thumb|500px|Figure 2. Schematic diagram of three screens in series.]]

[[File:VibratingScreenWhiten3.png|thumb|500px|Figure 3. Schematic diagram of three panel rows in series, analogous to the three screens in series of Figure 2.]]

The Whiten screen model combines particle classification with a relationship for screening across sequential rows of panels with differing aperture and open area properties.

=== Classification ===

This model applies a modified version of the Whiten and White (1977) expression for classification of particles to oversize:{{Whiten and White (1977)}}

{{Model theory (Text, Whiten and White Partition Curve)}}

=== Multi-row screening ===

{{Model theory (Text, Screen Multi-row)|Figure1=1|Figure2=2|Figure3=3}}

=== Screen efficiency ===

{{Model theory (Text, Screen Efficiency)}}

The efficiency of a given screen feed and deck configuration is a function of the <math>n_{\rm L}</math>, <math>R_{\rm f}</math> and <math>\lambda</math> model parameters.

Alternatively, the <math>n_{\rm L}</math> value required to achieve a target efficiency value for a given screen, feed and <math>R_{\rm f}</math>, <math>\lambda</math> parameters may be found by the model with a numerical root-finding algorithm.

=== Multiple decks ===

{{Model theory (Text, Screen Multiple Decks)}}

== Excel ==

The Whiten vibrating screen model may be invoked from the Excel formula bar with the following function call:

<syntaxhighlight lang="vb">=mdUnit_Screen_Whiten(Parameters as Range, Size as Range, Feed as Range, Rows as Range)</syntaxhighlight>

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>Parameters=
\begin{bmatrix}
\text{Method}\\
d_{\rm AR} \text{ (mm/mm)}\\
R_{\rm f} \text{ (frac)}\\
\lambda\\
(E_{\rm US})_{\rm T} \text{ (frac)}\\
n_{\rm L} \text{ (/m)}\\
\end{bmatrix},\;\;\;\;\;\;

Size = \begin{bmatrix}
d_{1}\text{ (mm)}\\ 
\vdots\\ 
d_n\text{ (mm)}\\ 
\end{bmatrix},\;\;\;\;\;\;

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},\;\;\;\;\;\;

Rows= \begin{bmatrix}
L_1\text{ (m)} & (D_1)_1\text{ (mm)} & (D_2)_1\text{ (mm)} & (f_{\rm o})_1\text{ (m}^2\text{/m}^2\text{)}\\ 
\vdots & \vdots & \vdots & \vdots\\ 
L_p\text{ (m)} & (D_1)_p\text{ (mm)} & (D_2)_p\text{ (mm)} & (f_{\rm o})_p\text{ (m}^2\text{/m}^2\text{)}\\ 
\end{bmatrix}\;\;\;\;\;\;
</math>


where:
* <math>\text{Method}</math> indicates whether the model applies a user-specified <math>n_L</math> parameter or solves the <math>n_L</math> value required to achieve a user-specified target efficiency (''0 = Specify n<sub>L</sub>, 1 = Specify Efficiency)''
* <math>(E_{\rm US})_{\rm T}</math> is the target screen efficiency (frac) if <math>\text{Method}=1</math>, ignored otherwise
* <math>n_{\rm L}</math> is number of trials parameter (/m) applied if <math>\text{Method}=0</math>, ignored otherwise
* <math>d_i</math> is the size of the square mesh interval that feed mass is retained on (mm)
* <math>m</math> is the number of ore types
* <math>Q_{\rm M,F}</math> is feed solids mass flow rate by size and ore type (t/h)

=== Results ===

The results are displayed in Excel as an array corresponding to the matrix notation below:

:<math>
mdUnit\_Screen\_Whiten = 
\begin{bmatrix}

\begin{array}{c}
\begin{bmatrix}
(E_{\rm US})_{\rm A}\\
n_{\rm L} \text{ (/m)}\\
\end{bmatrix}
&\\
&\\
\end{array}

&

\begin{bmatrix}
\bar d_1\text{ (mm)}\\
\vdots\\
\bar d_n\text{ (mm)}
\end{bmatrix}


&

\begin{bmatrix}
(Q_{\rm M,OS})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,OS})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,OS})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,OS})_{nm}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{bmatrix}
(Q_{\rm M,US})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,US})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,US})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,US})_{nm}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{bmatrix}
(E_{\rm oa})_1\text{ (frac)}\\
\vdots\\
(E_{\rm oa})_n\text{ (frac)}
\end{bmatrix}


\end{bmatrix}
</math>


where:
* <math>(E_{\rm US})_{\rm A}</math> is the actual efficiency of undersize removal achieved by the screen (frac)
* <math>n_{\rm L}</math> is number of trials parameter (/m) specified by the user if <math>\text{Method}=0</math>, or required to achieve the target efficiency if if <math>\text{Method}=1</math>
* <math>Q_{\rm M,OS}</math> is mass flow rate of solids to the oversize stream (t/h)
* <math>Q_{\rm M,US}</math> is mass flow rate of solids to the undersize stream (t/h)

=== Example ===

The images below show the selection of input arrays and output results in the Excel interface.

{| 
|- style="vertical-align:top;"
| 
[[File:VibratingScreenWhiten4.png|left|frame|Figure 4. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]] 
[[File:VibratingScreenWhiten6.png|left|frame|Figure 5. Example showing the selection of the '''Rows''' (green frame) array in Excel.]] 
| [[File:VibratingScreenWhiten5.png|left|frame|Figure 6. Example showing the selection of the '''Size''' (red frame), and '''Feed''' (purple frame) arrays in Excel.]]  
| [[File:VibratingScreenWhiten7.png|left|frame|Figure 7. Example showing the outline of the '''Results''' (light blue frame) array in Excel.]]
|}

== SysCAD ==

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

Note that both a '''Deck''' page and a '''Partition''' page are provided provided for each connected oversize discharge stream.

{{SysCAD (Page, Screen, DLL*Screen)}}

==== Deck page ====

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

{{SysCAD (Text, Table Header)}}

{{SysCAD (Text, Screen, Deck)}}

{{SysCAD (Text, Help Link)}}

|-
! colspan="3" style="text-align:left;" |''Parameters''
|- style="vertical-align:top;"
| rowspan="2" | Method
|Specify nL
|The user specifies the value of the nL parameter and efficiency is computed.
|- style="vertical-align:top;"
|Specify Efficiency
|The value of the nL parameter is adjusted to achieve a user-specified target efficiency value.
|-
|nL
|Input
|User specified value of the nL parameter. Only visible if the ''Specify nL'' method is selected above.
|-
|Efficiency
|Input
|User specified target value of screen efficiency. Only visible if ''Specify Efficiency'' method is selected above.
|-
|ParticleAspectRatio / ParticleAR
|Input
|Value of the particle aspect ratio, i.e. (second longest dimension):(longest dimension).
|-
|Rf
|Input
|Fines recovery parameter, fraction of the finest particles or liquids split to oversize stream.
|-
|Lambda
|Input
|Fines recovery parameter, shape factor.
|-
! colspan="3" style="text-align:left;" |''Rows''
|-
|NumRows
|Input
|Number of screen panel rows.
|-
|L
|Input
|Length of the row in the direction of travel.
|-
|ApertureL / D1
|Input
|Aperture length.
|-
|ApertureW / D2
|Input
|Aperture width.
|-
|OpenArea / fo
|Input
|Open area fraction of the row.

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

|-
! colspan="3" style="text-align:left;" |''Results''
|-
|Efficiency
|style="background: #eaecf0" | Display
|Actual efficiency of undersize removal achieved by the deck.
|-
|nL
|style="background: #eaecf0" | Display
|Value of nL used for the deck at the actual efficiency.
|}

{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Partition|ActionL=partition|DestinationU=Oversize|DestinationL=oversize|UnitL=screen}}

{{SysCAD (Page, About)}}

==== Additional notes ====

{{SysCAD (Text, No PSD Splits)|gasstream=undersize}}

== See also ==

* [[Partition (Size, Whiten and White)]]
* [[Vibrating Screen (Metso)]]

== References ==

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