Vibrating Screen (Karra)

Description

This article describes the Karra model for vibrating screen classification.

Model theory

Karra provides a method of computing the classification of particles by size on a vibrating screen.

Capacity

The capacity of a screen with particular properties is defined by the theoretical undersize transmission rate, $Th$ (t/h):

$Th=ABCDEFG_{c}\times {\text{Custom factor}}\times {\text{Screen area}}$

where:

• $A$ is the basic capacity of the screen (t/h/m²)
• $B$ is the oversize factor
• $C$ is the half-size factor
• $D$ is the deck location factor
• $E$ is the wet screening factor
• $F$ is the bulk density factor
• $G_{\rm {c}}$ is the near-size factor
• ${\text{Custom factor}}$ is a user defined factor
• ${\text{Screen area}}$ is the length x width of the screening area.

Basic capacity (A)

The basic capacity of a screen is defined by the throughfall aperture, $h_{\rm {T}}$ (mm), defined as the effective size of the aperture that a particle is presented to:

$h_{\rm {T}}=(h+d_{\rm {w}})\cos \theta -d_{\rm {w}}$

where:

• $h$ is the actual aperture of the screen (mm)
• $d_{\rm {w}}$ is the width of the wire or distance between apertures (mm)
• $\theta$ is the angle of inclination of the screen (deg.)

The basic capacity, $A$ (t/h/m²), is then:

$A={\begin{cases}12.13{h_{\rm {T}}}^{0.32}-10.3&h_{\rm {T}}<51{\text{ mm}}\\0.34{h_{\rm {T}}}+14.41&h_{\rm {T}}\geq 51{\text{ mm}}\\\end{cases}}$

The basic capacity equation assumes a woven wire mesh screening surface, and is corrected for other surfaces via the $OA$ factor:

${\rm {OA}}=21.5\log _{10}h+37$

The basic capacity computed above is then adjusted by the open area factor:

${\dfrac {A\times {\text{Open area fraction (}}\%{\text{)}}}{\rm {OA}}}$

where ${\text{Open area fraction}}$ (%) is the fraction of screening area consisting of apertures.

Oversize factor (B)

The oversize factor, $B$ is related to the fraction of screen fed which is larger than the throughfall aperture:

$B={\begin{cases}1.6-1.2(1-P_{h_{\rm {T}}})&(1-P_{h_{\rm {T}}})\leq 0.87\\4.275-4.25(1-P_{h_{\rm {T}}})&(1-P_{h_{\rm {T}}})>0.87\\\end{cases}}$

where $P_{h_{\rm {T}}}$ is the fraction of screen feed which is smaller than the throughfall aperture (frac).

Half-size factor (C)

The half-size factor, $C$ is related to the fraction of screen fed which is smaller than half the throughfall aperture size, $P_{h_{0.5{\rm {T}}}}$ (frac):

$C={\begin{cases}0.7+1.2P_{0.5h_{\rm {T}}}&P_{0.5h_{\rm {T}}}\leq 0.3\\2.053(P_{0.5h_{\rm {T}}})^{0.564}&0.3<P_{0.5h_{\rm {T}}}\leq 0.55\\3.35(P_{0.5h_{\rm {T}}})^{1.37}&0.55<P_{0.5h_{\rm {T}}}\leq 0.8\\5.0P_{0.5h_{\rm {T}}}-1.5&P_{0.5h_{\rm {T}}}>0.8\\\end{cases}}$

Deck location factor (D)

The deck location factor, $D$ is:

$D=1.1-0.1S$

where $S$ is the location of the deck, i.e. $S=1$ is the top deck, $S=2$ is the second deck etc.

Wet screening factor (E)

When spray water is added or slurry is screened, the wet screening factor, $E$, is:

$E={\begin{cases}1.0&T<1\\T&1\leq T<2\\1.5+0.25T&2\leq T<4\\2.5&4\leq T<6\\3.25-0.125T&6\leq T<10\\4.5-0.25T&10\leq T<12\\2.1-0.05T&12\leq T<16\\1.5-0.0125T&16\leq T<24\\1.35-0.00625T&24\leq T<32\\1.15&T\geq 32\\\end{cases}}$

where

$T=1.26h_{\rm {T}}$

For dry screening, $E=1.0$.

Bulk density factor (F)

The bulk density (or material weight) factor, $F$, is:

$F={\dfrac {\rho _{\rm {B}}}{1600}}$

where $\rho _{\rm {B}}$ is the bulk density of the feed (kg/m³). See Properties (Bulk Density) for methods to estimate bulk density based on particle size distribution.

Near-size capacity factor (G_c)

The near-size capacity factor, $G_{\rm {c}}$, is:

$G_{\rm {c}}=0.844{\big [}1-P_{1.25h_{\rm {T}}}-P_{0.75h_{\rm {T}}}){\big ]}^{3.453}$

where

• $P_{h_{1.25{\rm {T}}}}$ is the fraction of screen feed which is smaller than 1.25 times the throughfall aperture (frac), and
• $P_{h_{0.75{\rm {T}}}}$ is the fraction of screen feed which is smaller than 0.75 times the throughfall aperture (frac).

Classification

The partition of particles to screen oversize is:

$E_{{\rm {os}}i}=1-\exp \left[-0.693\left({\dfrac {{\bar {d}}_{i}}{d_{50}}}\right)^{5.9}\right]$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $E_{{\rm {os}}i}$ is the 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)

The value of $d_{50}$ (mm) is given by:

$d_{50}=h_{\rm {T}}K^{-0.148},\;\;\;\;\;\;K={\dfrac {(Q_{\rm {M,F}})_{P_{h_{\rm {T}}}}}{Th}}$

where $(Q_{\rm {M,F}})_{P_{h_{\rm {T}}}}$ is the mass flow rate of particles in the feed that are smaller than the throughfall aperture (t/h).

Load and efficiency

The screen load is defined as the actual rate of transmission to undersize as a fraction of the theoretical rate of transmission to undersize:

${\text{Load}}={\dfrac {Q_{\rm {M,US}}}{Th}}$

A load value less than 1.0 indicates an underloaded, oversized screen, and greater than 1.0 an overloaded, undersized screen.

A load value of 1.0 indicates the screen is sized exactly for the feed duty. Therefore, the screening area required to treat a given feed is:

${\text{Required area}}={\text{Load}}\times {\text{Screen area}}$

The screen efficiency is defined as the fraction of sub-aperture material in the feed that is actually recovered to undersize, i.e.:

${\text{Efficiency}}={\dfrac {Q_{\rm {M,US}}}{(Q_{\rm {M,F}})_{P_{h_{\rm {T}}}}}}$

Excel

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

=mdUnit_Screen_Karra(Parameters as Range, Size as Range, Feed 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:

$Parameters={\begin{bmatrix}\rho _{\rm {B}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\S\\\theta {\text{ (deg.)}}\\{\text{Wet screening (True/False)}}\\{\text{Screen length (m)}}\\{\text{Screen width (m)}}\\h{\text{ (mm)}}\\d_{\rm {w}}{\text{ (mm)}}\\{\text{ Open area fraction (frac)}}\\{\text{Custom factor}}\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;Feed={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}\;\;\;\;\;\;$

where:

• ${\text{Wet screening}}$ indicates whether the wet screening factor, $E$, should be applied (True/False)
• ${\text{Screen length}}$ is the length of the screening surface in the direction of flow (m)
• ${\text{Screen width}}$ is the width of the screening surface transverse to the direction of flow (m)
• $d_{i}$ 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:

$mdUnit\_Screen\_Karra={\begin{bmatrix}{\begin{bmatrix}{\text{Screen area (m}}^{2}{\text{)}}\\h_{\rm {T}}{\text{ (mm)}}\\A{\text{ (t/h/m}}^{2}{\text{)}}\\B\\C\\D\\T\\E\\F\\G_{\rm {c}}\\Th\\K\\(1-P_{h_{\rm {T}}}){\text{ (frac)}}\\P_{0.5h_{\rm {T}}}{\text{ (frac)}}\\(P_{1.25h_{\rm {T}}}-P_{0.75h_{\rm {T}}}){\text{ (frac)}}\\{\rm {OA}}{\text{ (frac)}}\\{\text{Required area (m}}^{2}{\text{)}}\\d_{50}{\text{ (mm)}}\\{\text{Load (frac)}}\\{\text{Efficiency (frac)}}\\\end{bmatrix}}{\begin{array}{ccccc}{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,OS}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,OS}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,OS}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,OS}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,US}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,US}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,US}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,US}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(E_{\rm {os}})_{1}{\text{ (frac)}}\\\vdots \\(E_{\rm {os}})_{n}{\text{ (frac)}}\end{bmatrix}}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\end{array}}\end{bmatrix}}$

where:

• $Q_{\rm {M,OS}}$ 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)
• $E_{\rm {os}}$ is partition fraction of feed solids to the oversize stream (frac)

Example

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

Figure 1. Example showing the selection of the Parameters (blue frame) array in Excel.

Figure 2. Example showing the selection of the Size (red frame), and Feed (purple frame) arrays in Excel.

Figure 4. 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
FeedBulkDensity / RhoBulk	Input	The bulk density of the feed solids.
DeckLocation / S	Input	Deck location, 1 = First, 2 = Second etc.
DeckInclination / Theta	Input	Angle of inclination of the deck.
WetScreening	Input	Indicates if the deck is wet screening (sprays, slurry etc).
ScreenLength	Input	Length of the screen.
ScreenWidth	Input	Width of the screen.
Aperture / h	Input	Size of the apertures in the deck.
WireDiameter / dw	Input	Diameter of the wire in the woven mesh screening surface (or distance between apertures).
OpenAreaFraction / ActualOA	Input	Open area fraction of the deck.
CustomFactor	Input	User defined factor for adjusting the theoretical screen capacity.
Liquids
LiquidsSeparMethod	Split To OS (User)	Liquids are split to oversize by a user-defined fraction of liquids in the feed.
	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
ScreenArea	Display	Area of the deck.
ThroughfallAperture/ hT	Display	Effective aperture size experienced by particles falling under gravity.
BasicCapacity / A	Display	Basic capacity of the screen.
OversizeFactor / B	Display	Particle oversize screen capacity factor.
HalfsizeFactor / C	Display	Particle halfsize screen capacity factor.
OversizeFactor / B	Display	Oversize screen capacity factor.
DeckLocationFactor / D	Display	Deck location screen capacity factor.
WetScreeningFactorT / T	Display	Wet screening screen capacity factor calculation parameter.
WetScreeningFactor / E	Display	Wet screening screen capacity factor.
NearSizeFactor / Gc	Display	Particle near-size screen capacity factor.
TheoreticalUndersize / Th	Display	Theoretical undersize transmission rate of the deck.
LoadingCoefficient / K	Display	Ratio of flow rate of sub aperture material in the feed to the theoretical screen capacity.
Halfsize	Display	Fraction of feed smaller than half the throughfall aperture.
Nearsize	Display	Fraction of feed near the throughfall aperture size.
Oversize	Display	Fraction of feed larger than the throughfall aperture.
OpenAreaWire / OA	Display	Estimated open area fraction of woven wire mesh.
RequiredArea	Display	Screening area required to treat feed stream at 100% loading.
d50	Display	Size at which the 50% of the feed particle mass is partitioned to oversize.
Load	Display	Ratio of mass flow of actual undersize to theoretical transmission rate to undersize of screen.
Efficiency	Display	Fraction of total sub-aperture sized material in feed that is actually recovered to the undersize stream.

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

• Vibrating Screen (Whiten)
• Vibrating Screen (Metso)