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).

Model theory

Figure 1. Example panel layout for a vibrating screen deck.

Figure 2. Schematic diagram of three screens in series.

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:

$E_{{\rm {oc}}i}={\begin{cases}\exp(-n_{L}Lf_{\rm {o}}{\dfrac {{\big [}D_{1}-{\bar {d}}_{i}{\big ]}{\big [}D_{2}-{\bar {d}}_{i}{\big ]}}{D_{1}D_{2}}})&{\bar {d}}_{i}<D_{2}\\1.0&{\bar {d}}_{i}\geq D_{2}\\\end{cases}}$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $E_{{\rm {oc}}i}$ is the corrected fraction of particles of size interval $i$ in the feed reporting to the oversize stream (frac)
• ${\bar {d}}_{i}$ is the geometric mean size of the size interval $i$ (mm)
• $n_{\rm {L}}$ is the number of trials per unit length parameter (/m)
• $L$ is the length of the screening area (or panel) in the direction of flow (m)
• $f_{\rm {o}}$ is the fraction open area of the screen deck/panels (m²/m²)
• $D_{1}$ is length (the longer side) of the screen deck/panel aperture (m)
• $D_{2}$ is width (the shorter side) of the screen deck/panel aperture (m)

Whiten and White's equation was modified by Dehghani et al. (2002) to include a term for irregularly shaped particles, and is generalised to:

$E_{{\rm {oc}}i}={\begin{cases}\exp(-n_{L}Lf_{\rm {o}}{\dfrac {{\big [}D_{1}-{\sqrt {2}}{\bar {d}}_{i}\cos \theta {\big ]}{\big [}D_{2}-{\sqrt {2}}{\bar {d}}_{i}\sin \theta {\big ]}}{D_{1}D_{2}}})&{\sqrt {2}}{\bar {d}}_{i}\sin \theta <D_{2}\\1.0&{\sqrt {2}}{\bar {d}}_{i}\sin \theta \geq D_{2}\\\end{cases}}$

where:

$\theta =\arctan d_{\rm {AR}}$

and $d_{\rm {AR}}$ is the representative aspect ratio of the particle population, the ratio of the second longest to the longest dimensions of a particle (mm/mm)

The aspect ratio property, $d_{\rm {AR}}$, allows for the balanced screening of 'flaky' or 'elongated' particles on slotted meshes. Dehghani et al.'s relation reduces to Whiten and White's original equation when ${\bar {d}}_{\rm {AR}}=1$.

Firth and Hart (2008) suggested a simple modification to a partition curve to account for the observed entrainment of fine particles in an oversize stream, with decreasing probability as particle size increases:

$E_{{\rm {oa}}i}=E_{{\rm {oc}}i}+\left(1-E_{{\rm {oc}}i}\right)\cdot R_{\rm {f}}\exp(-\lambda {\bar {d}}_{i})$

where:

• $E_{{\rm {oa}}i}$ is the actual fraction of particles of size interval $i$ in the feed reporting to the oversize stream (frac)
• $R_{\rm {f}}$ is the fraction of the finest particles or feed liquids split to the oversize stream (frac)
• $\lambda$ is the size constant

Multi-row screening

Figure 1 indicates a typical arrangement of screen panels on a screen deck. Each row of panels may exhibit a unique set of properties. Particles in the screen feed may therefore experience changing apertures sizes or open area fractions during screening, in the order of row arrangement in the flow direction.

Screening across the rows of a deck with differing properties is analogous to an arrangement of screens in series, where the oversize from the first screen (row) becomes the feed to the next screen (row), and so on. This concept is illustrated in Figures 2 and 3.

Mathematically, this is stated as:

$(E_{\rm {{oc}i}})_{\rm {Deck}}=(E_{\rm {{oc}i}})_{1}\times (E_{\rm {{oc}i}})_{2}\times \dots \times (E_{\rm {{oc}i}})_{p}=\prod _{k=1}^{p}{(E_{\rm {{oc}i}})_{k}}$

and adjusted for the entrainment of fines by:

$(E_{{\rm {oa}}i})_{\rm {Deck}}=(E_{{\rm {oc}}i})_{\rm {Deck}}+\left(1-(E_{{\rm {oc}}i})_{\rm {Deck}}\right)\cdot R_{\rm {f}}\exp(-\lambda {\bar {d}}_{i})$

where:

• $(E_{\rm {{oc}i}})_{\rm {Deck}}$ is the corrected fraction of the feed to the screen deck in size interval $i$ recovered to the deck oversize stream (frac)
• $(E_{\rm {{oc}i}})_{k}$ is the corrected fraction of the feed to panel row $k$ in size interval $i$ recovered to the row oversize stream (frac)
• $(E_{\rm {{oa}i}})_{\rm {Deck}}$ is the actual fraction of the feed to the screen deck in size interval $i$ recovered to the deck oversize stream (frac)
• $p$ is the number of panel rows with apertures permitting the passage of particles

Classification for each row is computed via the modified Whiten and White equation presented above. The same $n_{\rm {L}}$ parameter is applied to all rows as a simplification.

Screen efficiency

Screen efficiency, or the efficiency of undersize removal, $E_{\rm {US}}$ (frac), is defined as "the fraction of sub-aperture material in the feed stream which is actually recovered to the undersize stream". Formally,

$E_{\rm {US}}={\dfrac {Q_{\rm {M,US}}}{(Q_{\rm {M,F}}).P_{\rm {Aperture}}}}$

where:

• $Q_{\rm {M,US}}$ is the mass flow rate of particles in the undersize stream (t/h)
• $Q_{\rm {M,F}}$ is the mass flow rate of particles in the feed stream (t/h)
• $P_{\rm {Aperture}}$ is the mass fraction of particles in the feed which are smaller than the aperture size (w/w)

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

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

Multiple decks

Screens with multiple decks can be simulated by stacking model instances, i.e.:

• The undersize stream from the first deck becomes the feed to the next deck, and so on.
• Each deck yields an oversize stream, which can remain separated or combined into an overall oversize stream, depending on the screen and circuit configuration.

The overall efficiency of a multi-deck screen may be computed with the screen feed and final deck undersize streams in place of deck feed and deck undersize.

Excel

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

=mdUnit_Screen_Whiten(Parameters as Range, Size as Range, Feed as Range, Rows as Range)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

Inputs

The required inputs are defined below in matrix notation with elements corresponding to cells in Excel row ($i$) x column ($j$) format:

$Parameters={\begin{bmatrix}{\text{Method}}\\d_{\rm {AR}}{\text{ (mm/mm)}}\\R_{\rm {f}}{\text{ (frac)}}\\\lambda \\(E_{\rm {US}})_{\rm {T}}{\text{ (frac)}}\\n_{\rm {L}}{\text{ (/m)}}\\\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;Feed={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;Rows={\begin{bmatrix}L_{1}{\text{ (m)}}&(D_{1})_{1}{\text{ (mm)}}&(D_{2})_{1}{\text{ (mm)}}&(f_{\rm {o}})_{1}{\text{ (m}}^{2}{\text{/m}}^{2}{\text{)}}\\\vdots &\vdots &\vdots &\vdots \\L_{p}{\text{ (m)}}&(D_{1})_{p}{\text{ (mm)}}&(D_{2})_{p}{\text{ (mm)}}&(f_{\rm {o}})_{p}{\text{ (m}}^{2}{\text{/m}}^{2}{\text{)}}\\\end{bmatrix}}\;\;\;\;\;\;$

where:

• ${\text{Method}}$ indicates whether the model applies a user-specified $n_{L}$ parameter or solves the $n_{L}$ value required to achieve a user-specified target efficiency (0 = Specify n_L, 1 = Specify Efficiency)
• $(E_{\rm {US}})_{\rm {T}}$ is the target screen efficiency (frac) if ${\text{Method}}=1$, ignored otherwise
• $n_{\rm {L}}$ is number of trials parameter (/m) applied if ${\text{Method}}=0$, ignored otherwise
• $d_{i}$ is the size of the square mesh interval that feed mass is retained on (mm)
• $m$ is the number of ore types
• $Q_{\rm {M,F}}$ 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:

$mdUnit\_Screen\_Whiten={\begin{bmatrix}{\begin{array}{c}{\begin{bmatrix}(E_{\rm {US}})_{\rm {A}}\\n_{\rm {L}}{\text{ (/m)}}\\\end{bmatrix}}&\\&\\\end{array}}&{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,OS}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,OS}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,OS}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,OS}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,US}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,US}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,US}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,US}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(E_{\rm {oa}})_{1}{\text{ (frac)}}\\\vdots \\(E_{\rm {oa}})_{n}{\text{ (frac)}}\end{bmatrix}}\end{bmatrix}}$

where:

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

Figure 4. Example showing the selection of the Parameters (blue frame) array in Excel. Figure 5. Example showing the selection of the Rows (green frame) array in Excel.

Figure 6. Example showing the selection of the Size (red frame), and Feed (purple frame) arrays in Excel.

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.

MD_Screen page

The first tab page in the access window will have this name.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Tag	Display	This name tag may be modified with the change tag option.
Condition	Display	OK if no errors/warnings, otherwise lists errors/warnings.
ConditionCount	Display	The current number of errors/warnings. If condition is OK, returns 0.
GeneralDescription / GenDesc	Display	This is an automatically generated description for the unit. If the user has entered text in the 'EqpDesc' field on the Info tab (see below), this will be displayed here. If this field is blank, then SysCAD will display the unit class ID.
Requirements
On	CheckBox	This enables the unit. If this box is not checked, then the MassFracToOS option appears below.
MassFracToOS	Input	Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the overflow stream when the model is off.
NumParallelUnits	Input	The number of parallel, identical units to simulate: Feed is divided by the number of parallel units before being sent to the unit model. Unit model products are multiplied back by the same value and returned to the SysCAD product streams. All unit model result values are shown per parallel unit.
Options
ShowQFeed	CheckBox	QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
SizeForPassingFracCalc	Input	Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section.
FracForPassingSizeCalc	Input	Fraction passing for Size calculation. The fraction input here will be shown in the Stream Summary section.
Stream Summary
MassFlow / Qm	Display	The total mass flow in each stream.
SolidMassFlow / SQm	Display	The Solids mass flow in each stream.
LiquidMassFlow / LQm	Display	The Liquid mass flow in each stream.
VolFlow / Qv	Display	The total Volume flow in each stream.
Temperature / T	Display	The Temperature of each stream.
Density / Rho	Display	The Density of each stream.
SolidFrac / Sf	Display	The Solid Fraction in each stream.
LiquidFrac / Lf	Display	The Liquid Fraction in each stream.
Passing	Display	The mass fraction passing the user-specified size (in the field SizeForPassingFracCalc) in each stream.
Passes	Display	The user-specified (in the field FracForPassesSizeCalc) fraction of material in each stream will pass this size fraction.

Deck page

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

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Deck
On	Checkbox	This enables the deck. If off, the feed to this deck passes directly to the next deck (or undersize) without partition.
Method	Partition (User)	The partition to oversize for each size interval is defined by the user. Different values can be used for different solids.
	Partition (Reid-Plitt)	The partition to oversize for each size interval is defined by a Reid-Plitt efficiency curve. Different parameters can be used for different solids.
	Partition (Whiten-Beta)	The partition to oversize for each size interval is defined by a Whiten-Beta efficiency curve. Different parameters can be used for different solids.
	Vibrating (Karra)	The Karra vibrating screen model is used to determine the partition of solids to oversize and undersize for each size interval.
	Vibrating (Whiten)	The Whiten vibrating screen model is used to determine the partition of solids to oversize and undersize for each size interval.
	Vibrating (Metso)	The Metso vibrating screen model is used to determine the partition of solids to oversize and undersize for each size interval.
	Fine Wet (Mwale)	The Mwale fine wet screen model is used to determine the partition of solids to oversize and undersize for each size interval.
	Dewatering (Ng)	The Ng dewatering screen model is used to determine the moisture content of oversize material, and the partition of solids to oversize and undersize for each size interval.
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Parameters
Method	Specify nL	The user specifies the value of the nL parameter and efficiency is computed.
	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.
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.
Liquids
LiquidsSeparMethod	Split To OS (User)	Liquids are split to oversize by a user-defined fraction of liquids in the feed.
	Split To OS (Rf)	Liquids are split to oversize by the Rf fraction specified for the screen model.
	OS Solids Fraction	Sufficient liquids mass is recovered to the oversize stream to yield the user-defined oversize solids mass fraction value (if possible).
	OS Liquids Fraction	Sufficient liquids mass is recovered to the oversize stream to yield the user-defined oversize liquids mass fraction value (if possible).
OSSolidsFracReqd / OS.SfReqd	Input	Required value of the mass fraction of solids in the oversize stream. Only visible if OS Solids Fraction is selected.
OSLiquidsFracReqd / OS.LfReqd	Input	Required value of the mass fraction of liquids in the oversize stream. Only visible if OS Liquids Fraction is selected.
LiqSplitToOS / OS.LiqSplit	Input/Display	The fraction of feed liquids recovered to the oversize stream.
Results
Efficiency	Display	Actual efficiency of undersize removal achieved by the deck.
nL	Display	Value of nL used for the deck at the actual efficiency.

Partition page

The Partition page is used to specify or display the partition by species and size values.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Distribution
Name	Display	Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream.
IntervalCount	Display	Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream.
SpWithPSDCount	Display	Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
Partition
Method	Model/User	Select model-calculated or user-defined partition to separate each solids species type.
Density	Display	Density of each solid species.
Size	Display	Size of each interval in mesh series.
MeanSize	Display	Geometric mean size of each interval in mesh series.
All (All column)	Display	Actual overall partition to oversize of all solid species, for each size interval. Excludes solid species not present in the screen feed.
Partition	Display	Partition to oversize for each size interval, in each solid species, as determined by the selected model or user defined value. Note: These values are displayed regardless of whether the solid species is present in the screen feed or not.
All (All row, All column)	Display	Displays the actual, total, partition of all solids with a particle size distribution property in the feed to oversize. Excludes solid species not present in the screen feed.
All (All row, per species)	Display	Actual overall partition to oversize for each solid species, for all size intervals in that species. Excludes solid species not present in the screen feed.

About page

This page is provides product and licensing information about the Met Dynamics Models SysCAD Add-On.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
About
HelpLink		Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information		Copies Product and License information to the Windows clipboard.
Product
Name	Display	Met Dynamics software product name
Version	Display	Met Dynamics software product version number.
BuildDate	Display	Build date and time of the Met Dynamics Models SysCAD Add-On.
License
File		This is used to locate a Met Dynamics software license file.
Location	Display	Type of Met Dynamics software license or file name and path of license file.
SiteCode	Display	Unique machine identifier for license authorisation.
ReqdAuth	Display	Authorisation level required, MD-SysCAD Full or MD-SysCAD Runtime.
Status	Display	License status, LICENSE_OK indicates a valid license, other messages report licensing errors.
IssuedTo	Display	Only visible if Met Dynamics license file is used. Name of organisation/seat the license is authorised to.
ExpiryDate	Display	Only visible if Met Dynamics license file is used. License expiry date.
DaysLeft	Display	Only visible if Met Dynamics license file is used. Days left before the license expires.

Additional notes

• Solid species that do not possess a particle size distribution property are split according to the overall mass split of the default particle size distribution species selected in the SysCAD Project Configuration.
• If the default particle size distribution species is not present in the unit feed, the overall split of all other species with particle size distributions combined is used, as determined by the model.
• Gas phase species report directly to the undersize stream without split.

See also

• Partition (Size, Whiten and White)
• Vibrating Screen (Metso)