View source for Vibrating Screen (Metso)

== Description ==

This article describes an implementation of the vibrating screen sizing methodology described by '''Metso''' (Viilo, 2011), coupled with the classification methods of Whiten and White (1977), Dehghani et al. (2002), and Firth and Hart (2008).{{Viilo (2011)}}{{Whiten and White (1977)}}{{Dehghani et al. (2002)}}{{Firth and Hart (2008)}}

== Model theory ==

This model estimates the undersize transmission capacity of a given vibrating screen, from which the load and efficiency may be determined. The efficiency value is subsequently used to determine the classification performance of the screen, as outlined below.

=== Screen capacity ===

{| style="float:right; margin-left: 10px;"
|-
| [[File:VibratingScreenMetso1.png|thumb|425px|Figure 1. Specific screen capacity (after Viilo, 2011).{{Viilo (2011)}}]]
| [[File:VibratingScreenMetso2.png|thumb|425px|Figure 2. Retained material factor, B (after Viilo, 2011).{{Viilo (2011)}}]]
|-
| [[File:VibratingScreenMetso3.png|thumb|425px|Figure 3. Half-size factor, C (after Viilo, 2011).{{Viilo (2011)}}.]]
| [[File:VibratingScreenMetso4.png|thumb|425px|Figure 4. Wet screening factor, E (after Viilo, 2011).{{Viilo (2011)}}]]
|-
| [[File:VibratingScreenMetso5.png|thumb|425px|Figure 5. Moisture factor, L, (after Viilo, 2011).{{Viilo (2011)}}]]
| [[File:VibratingScreenMetso6.png|thumb|425px|Figure 6. Screen load-efficiency relationship (after Viilo, 2011).{{Viilo (2011)}}]]
|-
|colspan="2" | 
{| style="margin-left: auto; margin-right: 0px;"
|- style="vertical-align:top;"
| style="padding: 10px" |
{| class="wikitable"
|+ Table 1. Deck position (D) 
|-
! Deck position !! Factor (D)
|-
| First|| 1.0
|-
| Second|| 0.9
|-
| Third|| 0.8
|-
| Fourth|| 0.7
|} 
| style="padding: 10px" |
{| class="wikitable"
|+ Table 2. Shape of surface opening (H) 
|-
! Opening !! Factor (H)
|-
| Round || 0.9
|-
| Square || 1.0
|-
| Rectangular || 1.05
|}
| style="padding: 10px" |
{| class="wikitable"
|+ Table 3. Particle shape (I) 
|-
! Shape !! Factor (I)
|-
| Rounded || 1.2
|-
| Cubical || 1.0
|-
| Flaky|| 0.9
|}
| style="padding: 10px" |
{| class="wikitable"
|+ Table 4. Screen type (K)
|-
! Slope !! Deck !! Vibration !! Factor (K)
|-
|rowspan="2" | Horizontal
| Straight || Linear|| 0.9 
|-
| Straight || Constant elliptical|| 1.1
|-
|rowspan="3" | Inclined
| Straight || Circular || 1.0
|-
| Straight || Variable elliptical|| 1.1
|-
| Straight || Linear|| 1.0
|-
|rowspan="4" | Multislope
| Multislope|| Linear || 1.3
|-
| Triple slope  || Variable elliptical|| 1.4
|-
| Dual slope|| Linear || 1.1
|-
| Dual slope|| Variable elliptical|| 1.3
|}
|}
|}

[[File:VibratingScreenWhiten1.png|thumb|550px|Figure 7. Example panel layout for a vibrating screen deck.]]

[[File:VibratingScreenWhiten2.png|thumb|500px|Figure 8. Schematic diagram of three screens in series.]]

[[File:VibratingScreenWhiten3.png|thumb|500px|Figure 9. Schematic diagram of three panel rows in series, analogous to the three screens in series of Figure 8.]]

Viilo (2011) outlines a method for the sizing of vibrating screens for a given feed duty and desired separation.{{Viilo (2011)}}

The screen sizing method may be reversed for already existing screens and used to determine the nominal throughput capacity. The nominal capacity may be compared to the actual feed duty to generate a screen ''load'' performance parameter, from which the screen ''efficiency'' may be determined.

The ''specific screening capacity'', <math>Q_{\rm {spec}}</math> (t/h/m<sup>2</sup>) is defined as:

:<math>Q_{\rm {spec}} = A \cdot B \cdot C \cdot D \cdot E \cdot F \cdot G \cdot  H \cdot I \cdot J \cdot K \cdot L</math>

where:
* <math>A</math> is the specific screening capacity (t/h/m<sup>2)</sup>
* <math>B</math> is the retained (oversize) factor (-)
* <math>C</math> is the halfsize factor (-)
* <math>D</math> is the deck position factor (-)
* <math>E</math> is the wet screening factor (-)
* <math>F</math> is the material weight factor (-)
* <math>G</math> is the surface open area factor (-)
* <math>H</math> is the shape of the surface opening factor (-)
* <math>I</math> is the particle shape factor (-)
* <math>J</math> is the efficiency factor, which equals 1.0 for the reverse screening calculation (-)
* <math>K</math> is the screen type factor (-)
* <math>L</math> is the moisture factor (-)

The values of the factors <math>A</math>, <math>B</math>, <math>C</math>, <math>E</math>, and <math>L</math> are determined from the charts provided in Figures 1 - 5.

Factors <math>D</math>, <math>H</math>, <math>I</math> and <math>K</math> are similarly determined by consulting Tables 1 - 4.

Factor <math>F</math> is defined as:

:<math>F = \dfrac{\rho_{\rm S}}{2.7}</math>

where <math>\rho_{\rm S}</math> is the solid density (t/m<sup>3</sup>), and Factor <math>G</math> is:

:<math>G = \dfrac{\rm {OAF}}{0.5}</math>

where <math>\rm {OAF}</math> is the open area fraction of the screen deck (m<sup>2</sup>/m<sup>2</sup>).

The screen '''load''' (frac) is then defined as the "ratio of the flow rate of sub-aperture particles in the feed to the undersize transmission capacity of the screen", i.e.:

:<math>{\rm {Load}} = \dfrac{(Q_{\rm M, F}).P_{\rm Aperture}.S}{Q_{\rm {spec}}.l.w}</math>

where:
* <math>Q_{\rm {M,F}}</math> is the mass flow rate of particles in the feed stream (t/h)
* <math>P_{\rm {Aperture}}</math> is the mass fraction of particles in the feed which are smaller than the aperture size (w/w)
* <math>l</math> is the screen length in the direction of flow (m)
* <math>w</math> is the screen width in the transverse  direction to flow (m)
* <math>S</math> is the screen safety factor (-)

The safety factor, <math>S</math>, is typically 1.0, but may be adjusted to match performance data where some aspects of the screen design are unknown.

The target screen '''efficiency''' of undersize removal, <math>(E_{\rm US})_{\rm T}</math> (frac), may then be determined from the screen load via Figure 6.

A classification-by-size relationship is subsequently applied to the screen feed to yield the estimated screen efficiency.

=== 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=7|Figure2=8|Figure3=9}}

Note: For the purposes of screen capacity and efficiency calculations:
* The length of the screen (<math>l</math>) is the sum of the length of all defined rows.
* The aperture of the screen is defined as the smallest aperture dimension of all defined rows.
* The open area fraction of the screen is the row length-weighted average of the fraction open areas (<math>f_{\rm o}</math> of all defined rows.

=== Screen efficiency ===

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

The screen model automatically adjusts the <math>n_L</math> classification parameter to match <math>E_{\rm US}</math> to the target efficiency value, <math>(E_{\rm US})_{\rm T}</math>, determined by the screen capacity relationships above.

=== Multiple decks ===

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

== Excel ==

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

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

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>Parameters=
\begin{bmatrix}
\text{Moisture (}%\text{ w/w)}\\
\text{Screen type}\\
\text{Deck position}\\
\text{Wet screening (True/False)}\\
w\text{ (m)}\\
S\text{ (-)}\\
d_{\rm AR} \text{ (mm/mm)}\\
R_{\rm f} \text{ (frac)}\\
\lambda\\
\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},\;\;\;\;\;\;
</math>

:<math>
OreSG= \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},\;\;\;\;\;\;

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{Moisture}</math> is the mass fraction of moisture in the screen feed (% w/w)
* <math>\text{Screen type}</math> is the deck slope and motion characteristic of the screen:
** ''0 = Horizontal, Straight, Linear''
** ''1 = Horizontal, Straight, Const. Elliptical''
** ''2 = Inclined, Straight, Circular''
** ''3 = Inclined, Straight, Variable Elliptical''
** ''4 = Inclined, Straight, Linear''
** ''5 = Multislope, Linear''
** ''6 = Multislope, Triple slope, Variable Elliptical''
** ''7 = Multislope, Dual slope, Linear''
** ''8 = Multislope, Dual slope, Variable Elliptical''
** ''9 = Ignore''
* <math>\text{Wet screening}</math> indicates whether the wet screening factor, <math>E</math>, should be applied (''True/False'')
* <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\_Metso = 
\begin{bmatrix}

\begin{bmatrix}
l\text{ (m)}\\
\text{Aperture (mm)}\\
\text{OAF (m}^2\text{/m}^2\text{)}\\
\text{Opening shape}\\
\text{Particle shape}\\
\rho_{\rm S}\text{ (t/m}^3\text{)}\\
\text{Oversize fraction (frac)}\\
\text{Half-size fraction (frac)}\\
A\text{ (t/h/m}^2\text{)}\\
B\text{ (-)}\\
C\text{ (-)}\\
F\text{ (-)}\\
E\text{ (-)}\\
F\text{ (-)}\\
G\text{ (-)}\\
H\text{ (-)}\\
I\text{ (-)}\\
J\text{ (-)}\\
K\text{ (-)}\\
L\text{ (-)}\\
Q_{\rm spec}\text{ (-)}\\
Q_{\rm U}\text{ (-)}\\
\text{Screen area (m}^2\text{)}\\
\text{Required area (m}^2\text{)}\\
\text{Load (frac)}\\
(E_{\rm US})_{\rm T}\\
(E_{\rm US})_{\rm A}\\
n_{\rm L} \text{ (/m)}\\
\end{bmatrix}

\begin{array}{cccc}

\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}

\\
\\
\\
\\
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\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
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\\
\\
\\
\\
\\
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\\


\end{array}

\end{bmatrix}
</math>


where:
* <math>\text{Aperture}</math> is the smallest aperture dimension of all defined rows (mm)
* <math>\text{Opening shape}</math> is the shape of the apertures (''= 0 if all defined rows have square apertures, = 1 otherwise'')
* <math>\text{Particle shape}</math> is the shape of the feed particles based on the aspect ratio (''= 1 if <math>d_{\rm AR}=1</math>, = 2 otherwise'')
* <math>\text{Oversize fraction}</math> is the fraction of feed particles larger than the aperture size, i.e. <math>1-P_{\rm Aperture}</math> (frac)
* <math>\text{Half-size fraction}</math> is the fraction of feed particles smaller than the half the aperture size, i.e. <math>P_{0.5\rm Aperture}</math> (frac)
* <math>Q_{\rm U}</math> is the mass flow rate of particles in the feed smaller than the aperture size, i.e. <math>Q_{\rm M,F}.P_{\rm Aperture}</math> (frac)
* <math>\text{Screen area}</math> is the total area of all defined rows, i.e. <math>l \times w</math> (m<sup>2</sup>)
* <math>\text{Required area}</math> is the area required to screen the given feed at 100% load, i.e. <math>Q_{\rm U}.S \big/ Q_{\rm spec}</math> (m<sup>2</sup>)
* <math>(E_{\rm US})_{\rm T}</math> is the target efficiency of undersize removal required by the screen capacity calculations (frac)
* <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) required to achieve the target efficiency (/m)
* <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:VibratingScreenMetso7.png|left|frame|Figure 4. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]] 
[[File:VibratingScreenMetso9.png|left|frame|Figure 5. Example showing the selection of the '''Rows''' (pink frame) array in Excel.]] 
| [[File:VibratingScreenMetso8.png|left|frame|Figure 6. Example showing the selection of the '''Size''' (red frame), '''Feed''' (purple frame), and '''OreSG''' (green frame) arrays in Excel.]]  
| [[File:VibratingScreenMetso10.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 a '''Deck''' and '''Partition''' page is 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="10" | ScreenType
|Ignore
|The screen type factor (K) is ignored
|- style="vertical-align:top;"
|Horizontal, Straight, Linear
| rowspan="9" | The screen deck slope and motion configuration selected from Table 4.
|- style="vertical-align:top;"
|Horizontal, Straight, Const. Ellip.
|- style="vertical-align:top;"
|Inclined, Straight, Circular
|- style="vertical-align:top;"
|Inclined, Straight, Var. Ellip.
|- style="vertical-align:top;"
|Inclined, Straight, Linear
|- style="vertical-align:top;"
|Multislope, Linear
|- style="vertical-align:top;"
|Multislope, Triple slope, Var. Ellip.
|- style="vertical-align:top;"
|Multislope, Dual slope, Linear
|- style="vertical-align:top;"
|Multislope, Dual slope, Var. Ellip.
|-
|WetScreening
|True/False
|Indicates if the deck is wet screening (sprays, slurry etc).
|-
|DeckPosition
|Input
|Position of the deck, 1 = First, 2 = Second etc.
|-
|ScreenWidth / W
|Input
|Width of the screen.
|-
|SafetyFactor / S
|Input
|Value of the safety factor S.
|-
|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.
|-
|D1
|Input
|Aperture length.
|-
|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''
|-
|ScreenLength
|style="background: #eaecf0" | Display
|Total length of all defined rows.
|-
|Aperture
|style="background: #eaecf0" | Display
|Smallest aperture dimension of all defined rows.
|-
|OpenAreaFraction / OAF
|style="background: #eaecf0" | Display
|Average open area fraction of all defined rows.
|- style="vertical-align:top;"
|rowspan="3" | OpeningShape
|style="background: #eaecf0" | Round
|Aperture shape is round.
|-
|style="background: #eaecf0" | Square
|Aperture shape is square based on aperture dimensions of all rows.
|-
|style="background: #eaecf0" | Rectangular
|Aperture shape is rectangular based on aperture dimensions of all rows.
|- style="vertical-align:top;"
|rowspan="3" | ParticleShape
|style="background: #eaecf0" | Rounded
|Particle shape is rounded.
|-
|style="background: #eaecf0" | Cubical
|Particle shape is cubical based on particle aspect ratio.
|-
|style="background: #eaecf0" | Flaky
|Particle shape is flaky based on particle aspect ratio.
|-
|SolidsDensity / SRho
|style="background: #eaecf0" | Display
|Combined density of all solids with a particle size distribution property.
|-
|Oversize
|style="background: #eaecf0" | Display
|Fraction of oversize particles in deck feed.
|-
|Halfsize
|style="background: #eaecf0" | Display
|Fraction of half-size particles in deck feed.
|-
! colspan="3" style="text-align:left;" |''Factor''
|-
|BasicCapacity / A
|style="background: #eaecf0" | Display
|Basic capacity of the screen.
|-
|OversizeFactor / B
|style="background: #eaecf0" | Display
|Particle oversize factor.
|-
|HalfSizeFactor / C
|style="background: #eaecf0" | Display
|Particle half-size factor.
|-
|DeckPosition / D
|style="background: #eaecf0" | Display
|Deck location factor.
|-
|WetScreening / E
|style="background: #eaecf0" | Display
|Wet screening factor.
|-
|MaterialWeight / F
|style="background: #eaecf0" | Display
|Material weight (density) factor.
|-
|SurfaceOpenArea / G
|style="background: #eaecf0" | Display
|Open area factor.
|-
|OpeningShape / H
|style="background: #eaecf0" | Display
|Opening shape factor.
|-
|ParticleShape / I
|style="background: #eaecf0" | Display
|Particle shape factor.
|-
|Efficiency / J
|style="background: #eaecf0" | Display
|Efficiency factor.
|-
|ScreenType/ K
|style="background: #eaecf0" | Display
|Screen type factor.
|-
|Moisture / L
|style="background: #eaecf0" | Display
|Feed moisture factor.
|-
! colspan="3" style="text-align:left;" |''Load-Efficiency''
|-
|AvailableArea
|style="background: #eaecf0" | Display
|Area available for screening, i.e. ''total screen length x width''.
|-
|RequiredArea
|style="background: #eaecf0" | Display
|Screening area required to treat feed stream at 100% loading.
|-
|Load
|style="background: #eaecf0" | Display
|Ratio of the flow rate of sub-aperture particles in the deck feed to the undersize transmission capacity of the deck.
|-
|Efficiency
|style="background: #eaecf0" | Display
|Fraction of total sub-aperture sized material in deck feed that is actually recovered to the deck undersize stream.
|}

{{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}}

== References ==

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