View source for Air Classifier (Altun)

== Description ==

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

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

== Model theory ==

The Altun air classifier model applies the [[Partition (Size, Whiten-Beta)|Whiten-Beta efficiency curve]] to partition particles to the overflow stream:

{{Model theory (Text, Whiten-Beta Efficiency Curve)}}

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

=== Corrected cut size ===

The corrected cut size, <math>d_{50 \rm c}</math> (mm), is:

:<math>d_{50 \rm c} = k_{d_{50 \rm c}} \left ( \dfrac{\mathit{AF}}{ 3600 . \mathit{RS}.F} \right )^{n_{d_{50 \rm c}}}</math>

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

=== Sharpness of separation ===

The sharpness of separation term, <math>\alpha</math>, is estimated by:

:<math>\alpha = k_{\alpha} \left ( \dfrac{D}{\mathit{DL}} \right )^{n_{\alpha}}</math>

where:
* <math>D</math> is the classifier chamber diameter (m)
* <math>\mathit{DL}</math> is the dust loading of the classifier feed (kg/m<sup>3</sup>)
* <math>k_{\alpha}</math> and <math>n_{\alpha}</math> are the coefficient and exponent, respectively, of the sharpness of separation equation.

The dust loading, <math>\mathit{DL}</math> (kg/m<sup>3</sup>), is calculated from the feed and air flow rates:

:<math>\mathit{DL} = \dfrac{ 1000 . (Q_{\rm M,F})_{\rm S}}{\mathit{AF}}</math>

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

=== Fines bypass ===

The fraction of the finest particles reporting to overflow, <math>C</math> (frac), is:

:<math>C = \dfrac{100 - k_C (\mathit{DL})^{n_C}}{100}</math>

where <math>k_C</math> and <math>n_C</math> 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), <math>\beta</math>, is:

:<math>\beta = k_{\beta} \mathit{DL}^{n_{\beta}} - c_{\beta}</math>

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

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

Therefore, the <math>\beta^*</math> 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.

:{| class="wikitable"
|+ Table 1. Air classifier model equation parameters (after Altun and Benzer, 2014).{{Altun and Benzer (2014)}}
|-
! Equation !! Coefficient, <math>k</math> !! Exponent, <math>n</math> !! Constant, <math>c</math>
|-
| <math>\alpha</math> || 0.905 || 1.2679 || -
|-
| <math>d_{50 \rm c}</math> || 0.4048 || 0.7775 || -
|-
| <math>C</math> || 10.467 || 1.4171 || -
|-
| <math>\beta</math> || 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.{{Altun et al. (2016)}}{{Altun (2022)}}

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 ===

{{Model theory (Text, Hydrocyclone, Partition Metrics)}}

== Excel ==

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

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

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

=== Inputs ===

{{Excel (Text, Inputs)}}

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

</math>

where:
* <math>p</math> is the number of size intervals
* <math>m</math> is the number of ore types
* <math>d_i</math> is the size of the square mesh interval that mass is retained on (mm)
* <math>d_{i+1}<d_i<d_{i-1}</math>, i.e. descending size order from top size (<math>d_{1}</math>) to sub mesh (<math>d_{p}=0</math> mm)
* <math>(Q_{\rm M,F})_{\rm L}</math> 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:

:<math>
\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}
</math>


where:
* <math>(Q_{\rm M,OF})_{\rm L}</math> is the mass flow rate of liquids to the overflow stream (t/h)
* <math>(Q_{\rm M,UF})_{\rm L}</math> 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.

{|
|- style="vertical-align:top;"
| [[File:AirClassifierAltun1.png|left|frame|Figure 1. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]
| [[File:AirClassifierAltun2.png|left|frame|Figure 2. Example showing the selection of the '''Size''' (red frame), '''OreSG''' (green frame) and '''Feed''' (purple frame) arrays in Excel.]] 
| [[File:AirClassifierAltun3.png|left|frame|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.

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

==== Classifier page ====

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

{{SysCAD (Text, Table Header)}}

|-
! colspan="3" style="text-align:left;" |''Altun''

{{SysCAD (Text, Help Link)}}

|-
! colspan="3" style="text-align:left;" |''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.

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

|-
! colspan="3" style="text-align:left;" |''Results''
|-
|DustLoading / Feed.DL
|style="background: #eaecf0" | Display
|Dust loading of the feed.
|-
|FineSolidMassFlow / F
|style="background: #eaecf0" | Display
|Mass flow rate of -36+3 um solids in the feed.
|-
|Alpha
|style="background: #eaecf0" | Display
|Value of the ''alpha'' parameter of the Whiten Beta efficiency curve.
|-
|d50c
|style="background: #eaecf0" | Display
|Value of the ''d<sub>50c</sub>'' parameter of the Whiten Beta efficiency curve.
|-
|C
|style="background: #eaecf0" | Display
|Value of the ''C'' parameter of the Whiten Beta efficiency curve.
|-
|Beta
|style="background: #eaecf0" | Display
|Value of the ''beta'' parameter of the Whiten Beta efficiency curve.
|-
|BetaStar
|style="background: #eaecf0" | Display
|Value of the ''beta star'' parameter of the Whiten Beta efficiency curve.

{{SysCAD (Text, Hydrocyclone, Partition Metrics)}}

|}

{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Partition|ActionL=partition|DestinationU=Underflow|DestinationL=underflow|UnitL=classifier}}

{{SysCAD (Page, About)}}

==== Additional notes ====

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

== See also ==

* [[Partition (Size, Whiten-Beta)]]

== References ==

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