Air Classifier (Altun)

Description

This article describes the Altun and Benzer (2014) model for high efficiency air classifiers.

High efficiency air classifiers may be alternatively described as dynamic or third generation air classifiers.

Model theory

The Altun air classifier model applies the Whiten-Beta efficiency curve to partition particles to the overflow stream:

$E_{\rm {oa}}({\bar {d}}_{i})=C\left[{\dfrac {\left(1+\beta \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)(\exp(\alpha )-1)}{\exp \left(\alpha \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)+\exp(\alpha )-2}}\right]$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $E_{\rm {oa}}({\bar {d}}_{i})$ is the fraction of particles of size interval $i$ in the feed reporting to the overflow stream (frac)
• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)
• $d_{\rm {50c}}$ is the corrected size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
• $C$ is the fraction of feed liquids (or fines) split to overflow (frac)
• $\alpha$ is a parameter representing the sharpness of separation
• $\beta$ is a term introduced to accommodate the so-called fish-hook effect, and controls the initial rise in the efficiency curve at finer sizes
• $\beta ^{*}$ is computed to ensure the Whiten-Beta function preserves the definition of $d_{\rm {50c}}$ in the presence of the fish-hook, i.e. $E=0.5C$ at $d_{\rm {50c}}$

Additional relations link terms of the efficiency curve equation with operating parameters.

Corrected cut size

The corrected cut size, $d_{50{\rm {c}}}$ (mm), is:

$d_{50{\rm {c}}}=k_{d_{50{\rm {c}}}}\left({\dfrac {\mathit {AF}}{3600.{\mathit {RS}}.F}}\right)^{n_{d_{50{\rm {c}}}}}$

where:

• ${\mathit {AF}}$ is the classifier air flow rate (m³/h)
• ${\mathit {RS}}$ is the rotor speed of the classifier (m/s)
• $F$ is the flow rate of -36+3 µm size particles in the feed (t/h)
• $k_{d_{50{\rm {c}}}}$ and $n_{d_{50{\rm {c}}}}$ are the coefficient and exponent, respectively, of the cut size equation.

Sharpness of separation

The sharpness of separation term, $\alpha$, is estimated by:

$\alpha =k_{\alpha }\left({\dfrac {D}{\mathit {DL}}}\right)^{n_{\alpha }}$

where:

• $D$ is the classifier chamber diameter (m)
• ${\mathit {DL}}$ is the dust loading of the classifier feed (kg/m³)
• $k_{\alpha }$ and $n_{\alpha }$ are the coefficient and exponent, respectively, of the sharpness of separation equation.

The dust loading, ${\mathit {DL}}$ (kg/m³), is calculated from the feed and air flow rates:

${\mathit {DL}}={\dfrac {1000.(Q_{\rm {M,F}})_{\rm {S}}}{\mathit {AF}}}$

where $(Q_{\rm {M,F}})_{\rm {S}}$ is the mass flow rate of solids in the feed (t/h)

Fines bypass

The fraction of the finest particles reporting to overflow, $C$ (frac), is:

$C={\dfrac {100-k_{C}({\mathit {DL}})^{n_{C}}}{100}}$

where $k_{C}$ and $n_{C}$ are the coefficient and exponent, respectively, of the fines bypass equation.

Fish-hook parameter

The parameter controlling the initial rise of the efficiency curve in fines sizes (fish-hook), $\beta$, is:

$\beta =k_{\beta }{\mathit {DL}}^{n_{\beta }}-c_{\beta }$

where $k_{\beta }$, $n_{\beta }$ and $c_{\beta }$ are the coefficient, exponent and constant, respectively, of the fish-hook equation.

Altun and Benzer (2014) also provide an equation for the $\beta ^{*}$ term of the efficiency equation. However, Whiten's definition of $\beta ^{*}$ requires its value to be computed in complement to the specified value of $\beta$ in the efficiency equation (see Partition (Size, Whiten-Beta) for more information).

Therefore, the $\beta ^{*}$ equation is excluded from this implementation of Altun and Benzer's (2014) air classifier model.

Equation parameters

Altun and Benzer's (2014) test work was conducted on a range of Sepol, Sepax and Sepmaster industrial scale dynamic air classifiers. The parameter values in Table 1 were derived by regression analyses of their industrial sampling results.

Table 1. Air classifier model equation parameters (after Altun and Benzer, 2014).

Equation	Coefficient, $k$	Exponent, $n$	Constant, $c$
$\alpha$	0.905	1.2679	-
$d_{50{\rm {c}}}$	0.4048	0.7775	-
$C$	10.467	1.4171	-
$\beta$	0.4417	1.4171	0.1293

This implementation of the air classifier model allows a user to specify the values of the equation coefficients, exponents and constants directly. Altun and Benzer's (2014) values may be used, or new values derived from specific industrial or laboratory test work programmes.

Multi-component modelling

Altun and co-workers have published subsequent journal articles outlining similar model treatments for multi-component feeds, where particle density differences and agglomeration are observed to affect classification performance.

However, this work was only conducted on laboratory scale equipment rather than operating industrial air classifiers. Additional non-standard material measurements are also required (e.g. fluidity index). As such, these model implementations are not included here.

Partition metrics

Several metrics are provided to characterise the partition curve.

The $d_{50}$, also known as the cut or separation size, is defined as the size of a particle which has an even (50%) chance of appearing in either the underflow or overflow stream. The $d_{50}$ size is estimated via a log-linear interpolation of geometric mean size ($\ln {\bar {d}}$) against the uncorrected partition to underflow of all solids in the feed.

The Ecart Probable, or $E_{\rm {p}}$, is a measure of the deviation of a partition curve from a perfect separation, and is typically defined for size classification as:

$E_{\rm {p}}={\dfrac {d_{75}-d_{25}}{2}}$

where $d_{75}$ and $d_{25}$ are the sizes of particles which have a 75% and 25% probability, respectively, of appearing in the underflow stream. The $d_{75}$ and $d_{25}$ sizes are estimated by log-linear interpolation of geometric mean size against the uncorrected partition to underflow of all solids in the feed.

The Imperfection, $I$, is a normalised measure of the sharpness of separation, which is suggested to be independent of the magnitude of the $d_{50}$, and is typically defined for size classification as:

$I={\dfrac {E_{\rm {p}}}{d_{50}}}={\dfrac {d_{75}-d_{25}}{2d_{50}}}$

Excel

The Altun air classifier model may be invoked from the Excel formula bar with the following function call:

=mdUnit_AirClassifier_Altun(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:

$Parameters={\begin{bmatrix}D{\text{ (m)}}\\{\mathit {AF}}{\text{ (m}}^{3}{\text{/h)}}\\{\mathit {RS}}{\text{ (m/s)}}\\k_{\alpha }\\n_{\alpha }\\k_{d_{50}}\\n_{d_{50}}\\k_{C}\\n_{C}\\k_{\beta }\\n_{\beta }\\c_{\beta }\\(Q_{\rm {M,F}})_{\rm {L}}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{p}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;Feed={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{p1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{pm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;$

where:

• $p$ is the number of size intervals
• $m$ is the number of ore types
• $d_{i}$ is the size of the square mesh interval that mass is retained on (mm)
• $d_{i+1}<d_{i}<d_{i-1}$, i.e. descending size order from top size ($d_{1}$) to sub mesh ($d_{p}=0$ mm)
• $(Q_{\rm {M,F}})_{\rm {L}}$ is the mass flow rate of liquids in the feed (t/h)

Results

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

${\mathit {mdUnit\_AirClassifier\_Altun}}={\begin{bmatrix}{\begin{bmatrix}{\mathit {DL}}{\text{ (kg/m}}^{3}{\text{)}}\\F{\text{ (t/h)}}\\\alpha \\d_{\rm {50c}}{\text{ (mm)}}\\C{\text{ (frac)}}\\\beta \\\beta ^{*}\\(Q_{\rm {M,OF}})_{\rm {L}}{\text{ (t/h)}}\\(Q_{\rm {M,UF}})_{\rm {L}}{\text{ (t/h)}}\\d_{50}{\text{ (mm)}}\\E_{\rm {p}}{\text{ (mm)}}\\I\\\end{bmatrix}}{\begin{array}{cccccc}{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{p}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,UF}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,UF}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,UF}})_{p1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,UF}})_{pm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,OF}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,OF}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,OF}})_{p1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,OF}})_{pm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(E_{\rm {oa}})_{1}{\text{ (frac)}}\\\vdots \\(E_{\rm {oa}})_{p}{\text{ (frac)}}\end{bmatrix}}\\\\\\\\\\\\\\\end{array}}\end{bmatrix}}$

where:

• $(Q_{\rm {M,OF}})_{\rm {L}}$ is the mass flow rate of liquids to the overflow stream (t/h)
• $(Q_{\rm {M,UF}})_{\rm {L}}$ is the mass flow rate of liquids to the underflow stream (t/h)

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), OreSG (green frame) and Feed (purple frame) arrays in Excel.

Figure 3. Example showing the selection 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.

MD_Classifier 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 MassFracToUF option appears below.
MassFracToUF	Input	Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the underflow stream when the model is off.
Method	Partition (User)	The partition to underflow for each size interval is defined by the user. Different values can be used for different solids.
	Partition (Reid-Plitt)	The partition to underflow 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 underflow for each size interval is defined by a Whiten-Beta efficiency curve. Different parameters can be used for different solids.
	Air Classifier (Altun)	The Altun air classifier model is used to determine the partition of solids to underflow for each size interval.
	Gravity (Plitt-Flintoff)	The Plitt-Flintoff gravity classifier model is used to determine the partition of solids to underflow for each size interval.
Options
ShowQFeed	CheckBox	QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
ShowQOF	CheckBox	QOF and associated tab pages (eg Sp) will become visible, showing the properties of the overflow stream.
ShowQUF	CheckBox	QUF and associated tab pages (eg Sp) will become visible, showing the properties of the underflow 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.

Classifier page

The Classifier page is used to specify the input parameters for the classifier model.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Altun
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Parameters
Diameter / D	Input	Chamber diameter
AirFlowRate / AF	Input	Flow rate of air into chamber.
RotorSpeed / RS	Input	Velocity of the rotor.
Alpha	Input	Coefficient (k) and exponent (n) of the alpha equation.
d50c	Input	Coefficient (k) and exponent (n) of the d50c equation.
Beta	Input	Coefficient (k), exponent (n) and constant (c) of the beta equation.
Liquids
LiquidsSeparMethod	Split To UF (User)	Liquids are split to underflow by a user-defined fraction of liquids in the feed.
	Split To UF (1-C)	Liquids are split to underflow by the C (overflow) fraction specified for the first species in the model.
	UF Solids Fraction	Sufficient liquids mass is recovered to the underflow stream to yield the user-defined underflow solids mass fraction value (if possible).
	UF Liquids Fraction	Sufficient liquids mass is recovered to the underflow stream to yield the user-defined underflow liquids mass fraction value (if possible).
UFSolidsFracReqd / UF.SfReqd	Input	Required value of the mass fraction of solids in the underflow stream. Only visible if UF Solids Fraction is selected.
UFLiquidsFracReqd / UF.LfReqd	Input	Required value of the mass fraction of liquids in the underflow stream. Only visible if UF Liquids Fraction is selected.
LiqSplitToUF / UF.LiqSplit	Input/Display	The fraction of liquids recovered to underflow.
Results
DustLoading / Feed.DL	Display	Dust loading of the feed.
FineSolidMassFlow / F	Display	Mass flow rate of -36+3 um solids in the feed.
Alpha	Display	Value of the alpha parameter of the Whiten Beta efficiency curve.
d50c	Display	Value of the d50c parameter of the Whiten Beta efficiency curve.
C	Display	Value of the C parameter of the Whiten Beta efficiency curve.
Beta	Display	Value of the beta parameter of the Whiten Beta efficiency curve.
BetaStar	Display	Value of the beta star parameter of the Whiten Beta efficiency curve.
d50	Display	The separation size, d50, of all solid particles in the feed.
EcartProbable / Ep	Display	The value of the Ecart Probable of the separation.
Imperfection / I	Display	The value of the Imperfection of the separation.

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 underflow of all solid species, for each size interval. Excludes solid species not present in the classifier feed.
Partition	Display	Partition to underflow 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 classifier 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 underflow. Excludes solid species not present in the classifier feed.
All (All row, per species)	Display	Actual overall partition to underflow for each solid species, for all size intervals in that species. Excludes solid species not present in the classifier 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 overflow stream without split.

See also

• Partition (Size, Whiten-Beta)