View source for Fine Wet Screen (Mwale)

== Description ==

This article describes an implementation of the Mwale et al. (2016) model for fine wet screening.{{Mwale et al. (2016)}}

== Model theory ==

=== Efficiency curve ===

Mwale et al. (2016) proposed a phenomenological model of fine wet screening where the partition to oversize, <math>E_{i{\rm o}}</math> (frac), is related to properties of both the screen and the feed:{{Mwale et al. (2016)}}

:<math>E_{i{\rm o}} = \exp \left [ - \dfrac{A_{\rm o} K}{F_{\rm f} s \left ( \dfrac{\bar d_i}{a} \right)^\alpha} \right ] + \dfrac{\delta . F_{\rm f}}{100(1-s)} \left [ \exp \left ( - \dfrac{\bar d_i}{a} \right ) \right ]^\alpha</math>

where:
* <math>A_{\rm o}</math> is the total screen open area (m<sup>2</sup>)
* <math>F_{\rm f}</math> is the solids feed rate (t/h)
* <math>s</math> is the mass fraction of solids in the feed (w/w)
* <math>x_{\rm a}</math> is the aperture size (mm)
* <math>\bar d_{i}</math> is the [[Conversions|geometric mean size]] of the size interval <math>i</math> (mm)
* <math>K</math> is the kinetic constant (t/hr.m<sup>2</sup>)
* <math>\alpha</math> is the sharpness of separation parameter (-)
* <math>\delta</math> is the fines bypass parameter (h/t)

=== Sharpness constant ===

Mwale et al. (2018) suggest the following equation to estimate the sharpness constant, <math>\alpha</math>, as a function of screen design and feed conditions:{{Mwale et al. (2018)}}

:<math>\alpha = k_1 \left ( \dfrac{{\rho_{\rm p}}^2 F_{\rm f} {A_{\rm o}}^{3.5}}{\mu a^2 {M_{\rm u}}^2} \right ) + k_2 \left ( \dfrac{A_{\rm o}}{a^2} \right ) + k_3</math>

where:
* <math>\rho_{\rm p}</math> is the density of the feed slurry (t/m<sup>3</sup>)
* <math>M_{\rm u}</math> is the mass flow rate of particles in the feed which are smaller than the aperture (t/h)
* <math>k_1</math>, <math>k_2</math> and <math>k_3</math> are coefficients of the sharpness equation 

The apparent viscosity of slurry in the feed, <math>\mu</math> (Pa.s), is estimated by the classical Krieger-Dougherty (1959) equation:{{Krieger and Dougherty (1959)}}

:<math>\mu = \mu_{\rm 0} \left ( 1 - \dfrac{\phi_{\rm Eff}}{\phi_{\rm m}} \right)^{-k \phi_{\rm m}}</math>

where:
* <math>\mu_0</math> is the viscosity of the carrier fluid, i.e. water (Pa.s)
* <math>\phi_{\rm m}</math> is maximum packing fraction of particles (v/v), assumed to be 0.64
* <math>k</math> is a particle shape constant, assumed to be 2.5

The sharpness constant can be specified directly as the value of coefficient <math>k_3</math> when coefficients <math>k_1</math> and <math>k_2</math> are set to zero.

=== Bypass constant ===

Mwale et al. (2018) suggest the following simple linear relationship to estimate the bypass constant, <math>\delta</math>,:{{Mwale et al. (2018)}}

:<math>\delta = C_1.F_{\rm f} + C_2.s + C_3.x_{\rm a} + C_4</math>

where <math>C_1-C_4</math> are coefficients of the bypass equation.

The bypass constant can be specified directly as the value of coefficient <math>C_4</math> when coefficients <math>C_1-C_3</math> are set to zero.

== Excel ==

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

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

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>Parameters=
\begin{bmatrix}
x_{\rm a} \text{ (mm)}\\
A_{\rm o}\text{ (m}^2\text{)}\\
K\text{ (t/h.m}^2\text{)}\\
k_1\text{ (-)}\\
k_2\text{ (-)}\\
k_3\text{ (-)}\\
C_1\text{ (-)}\\
C_2\text{ (-)}\\
C_3\text{ (-)}\\
C_4\text{ (-)}\\
(Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\
\rho_{\rm L}\text{ (t/m}^{\text{3}}\text{)}\\
\mu_0\text{ (mPa.S)}\\
\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},\;\;\;\;\;\;

\mathit{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}

</math>


where:
* <math>n</math> is the number of size intervals
* <math>d_i</math> is the size of the square mesh interval that feed 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_{n}=0</math> 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)
* <math>(Q_{\rm M,F})_{\rm L}</math> is the mass flow feed rate of liquids in the feed (t/h)
* <math>\rho_{\rm S}</math> is the density of solids in the feed (t/m<sup>3</sup>)
* <math>\rho_{\rm L}</math> is the density of liquids in the feed (t/m<sup>3</sup>)

=== Results ===

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

:<math>
\mathit{mdUnit\_Screen\_Mwale} = 
\begin{bmatrix}

\begin{array}{c}
\begin{bmatrix}
\alpha\text{ (-)}\\
\delta\text{ ((t/h)}^{-1}\text{)}\\
F_{\rm f}\text{ (t/h)}\\
s\text{ (w/w)}\\
\rho_{\rm p}\text{ (t/m}^3\text{)}\\
\mu\text{ (Pa.s)}\\
M_{\rm u}\text{ (t/h)}\\
E_{\rm US}\text{ (frac)}\\
\end{bmatrix}
\end{array}

\begin{array}{cccccc}

\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_{i \rm o})_1\text{ (frac)}\\
\vdots\\
(E_{i \rm o})_n\text{ (frac)}
\end{bmatrix}

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

\end{array}
\end{bmatrix}
</math>


where:
* <math>E_{\rm US}</math> is the efficiency of undersize removal achieved by the screen (frac)
* <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:FineWetScreenMwale1.png|left|frame|Figure 1. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]] 
| [[File:FineWetScreenMwale2.png|left|frame|Figure 2. Example showing the selection of the '''Size''' (red frame), '''Feed''' (purple frame) and '''OreSG''' (green frame) arrays in Excel.]]  
| [[File:FineWetScreenMwale3.png|left|frame|Figure 3. 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''
|-
|Aperture / xa
|Input
|Size of the apertures in the deck.
|-
|OpenArea/ Ao
|Input
|Open area of the screening surface
|-
|KineticParameter / K
|Input
|Mwale model kinetic parameter
|-
|SharpnessCoeff1 / k1
|Input
|Mwale model sharpness coefficient
|-
|SharpnessCoeff2 / k2
|Input
|Mwale model sharpness coefficient
|-
|SharpnessCoeff3 / k3
|Input
|Mwale model sharpness coefficient
|-
|BypassCoeff1 / C1
|Input
|Mwale model bypass coefficient
|-
|BypassCoeff2 / C2
|Input
|Mwale mode3 bypass coefficient
|-
|BypassCoeff3 / C3
|Input
|Mwale model bypass coefficient
|-
|BypassCoeff4 / C4
|Input
|Mwale model bypass coefficient

{{SysCAD (Text, Screen, Liquids)|method=0}}

|-
! colspan="3" style="text-align:left;" |''Results''
|-
|SharpnessConstant / Alpha
|style="background: #eaecf0" | Display
|Mwale model sharpness constant
|-
|BypassConstant / Delta
|style="background: #eaecf0" | Display
|Mwale model bypass constant
|-
|Feed.SQm
|style="background: #eaecf0" | Display
|Mass flow rate of solids in the feed (excludes solids without PSD quality)
|-
|Feed.SQmSubAp
|style="background: #eaecf0" | Display
|Mass flow rate of particles in the feed which are smaller than the aperture size (sub-aperture)
|-
|Feed.Sf
|style="background: #eaecf0" | Display
|Mass fraction of solids in the feed (excludes solids without PSD quality)
|-
|Feed.SLRho
|style="background: #eaecf0" | Display
|Density of slurry in the feed
|-
|SlurryViscosity / mu
|style="background: #eaecf0" | Display
|Apparent viscosity of slurry in the feed
|-
|Efficiency
|style="background: #eaecf0" | Display
|Fraction of total sub-aperture sized material in feed that is actually recovered to the 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]]