View source for Dense Medium Drum (Baguley)

== Description ==

This article describes the Baguley and Napier-Munn (1996) model for dense medium drums or bath-type separators.{{Baguley and Napier-Munn (1996)}}

== Model theory ==

[[File:DenseMediumBaguley4.png|thumb|450px|Figure 1. Example of the partition curves generated by the Baguley dense medium drum model (lines) for an industrial data set (markers) (after Baguley, 1988.{{Baguley (1988)}}). Chart series represent density classes.]]

[[File:DenseMediumBaguley5.png|thumb|450px|Figure 2. Example of the partition curves generated by the Baguley dense medium drum model (lines) for the same industrial data set (markers) as Figure 1 (after Baguley, 1988.{{Baguley (1988)}}). Chart series represent size intervals.]]

Baguley and Napier-Munn (1996) proposed a model of dense-medium drum separators where the partitioning behaviour is related to the hydrodynamic properties of particles, such as terminal velocity in a fluid.{{Baguley and Napier-Munn (1996)}}

An empirical function is adopted for the fraction of feed particles reporting to the sinks stream, <math>P</math> (frac):

:<math>
P_{ij} =
\begin{cases}
\left [ 1 - (V_{100} - V_{{\rm t}_{ij}})^2 \right]^x & V_{\rm t} < V_{100}\\
1 & V_{\rm t} \geq V_{100}
\end{cases}
</math>

where particle size effects are incorporated via the <math>x</math> term:

:<math>\ln(x) = \dfrac{A}{{\bar d_i}^2} + B</math>

and:
* <math>i</math> is the index of the size interval, <math>i = \{1,2,\dots,q\}</math>, <math>q</math> is the number of size intervals
* <math>j</math> is the index of the density class, <math>j = \{1,2,\dots,m\}</math>, <math>m</math> is the number of density classes
* <math>V_{100}</math> is the terminal velocity of particles of which 100% report to sinks (m/s)
* <math>V_{\rm t}</math> is the terminal velocity of particles (m/s)
* <math>A</math> is a coefficient of the size expression (-)
* <math>B</math> is constant of the size expression (-)
* <math>\bar d_{i}</math> is the [[Conversions|geometric mean size]] of particles in size interval <math>i</math> (mm)

In order to compute the terminal velocity, the series of expressions described below are used.

The dimensionless terminal velocity, <math>V_{\rm t}^* </math>, is computed for each particle size (<math>i</math>) and density class (<math>j</math>) from:

:<math>V_{\rm t}^* = \dfrac{{\delta_0}^2}{4d^*} \left [ \left ( 1 + \dfrac{4{d^*}^{1.5}}{{C_0}^{0.5} {\delta_0}^2} \right )^{0.5} - 1 \right ]^2</math>

where <math>d^*</math> is the dimensionless particle size computed from:

:<math>d^* = \dfrac{\bar d}{p}</math>

and <math>\bar d</math> is the [[Conversions|geometric mean size]] of particles in a size interval (with units of m in the <math>d^*</math> equation).

The particle-fluid characteristic size parameter, <math>p</math>, is:

:<math>p = \left ( \dfrac{3 \mu^2}{4 (\Delta \rho) \rho_{\rm m} g} \right )^{\frac{1}{3}}</math>

where:
* <math>\mu</math> is the apparent equivalent Newtonian medium viscosity (Pa.s)
* <math>\Delta\rho</math> is density difference between a particle and the medium (kg/m<sup>3</sup>)
* <math>\rho_{\rm m}</math> is medium density (kg/m<sup>3</sup>)
* <math>g</math> is acceleration due to gravity (m/s<sup>2</sup>)

The shape-dependent parameters <math>C_0</math> and <math>\delta_0</math> are calculated from:

:<math>\ln C_0 = -5.99k + 1.95</math>

:<math>\delta_0 = 9.97k + 2.94</math>

where <math>k</math> is a particle shape coefficient, assumed to be 0.225 for spheres.

Finally, the particle terminal velocity, <math>v_{\rm t}</math> (m/s), is computed from:

:<math>V_{\rm t} = V_{\rm t}^* q</math>

where the particle-fluid characteristic velocity parameter, <math>q</math>, is:

:<math>q = \left ( \dfrac{4 (\Delta \rho) \mu g}{3 {\rho_{\rm m}}^2} \right )^{\frac{1}{3}}</math>

== Excel ==

The Baguley dense medium drum model may be invoked from the Excel formula bar with the following function call:

<syntaxhighlight lang="vb">=mdUnit_DMDrum_Baguley(Parameters as Range, Size as Range, Density As Range, Feed as Range</syntaxhighlight>

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>\mathit{Parameters} =
\begin{bmatrix}
\rho_{m}\text{ (t/m}^{\text{3}}\text{)}\\
\mu\text{ (mPa.s)}\\
k\\
V_{100}\text{ (m/s)}\\
A\\
B\\
\end{bmatrix},\;\;\;\;\;\;

\mathit{Size} = \begin{bmatrix}
d_{1}\text{ (mm)}\\ 
\vdots\\ 
d_n\text{ (mm)}\\ 
\end{bmatrix},\;\;\;\;\;\;

\mathit{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{Density}= \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>m</math> is the number of density classes
* <math>d_i</math> is the size of the square mesh interval that feed mass is retained on (mm)
* <math>Q_{\rm M,F}</math> is feed solids mass flow rate by size and ore type (t/h)
* <math>\rho_{\rm S}</math> is the particle density (t/m<sup>3</sup>)

=== Results ===

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

:<math>
\mathit{mdUnit\_DenseMedium\_Baguley} = 
\begin{bmatrix}

\begin{array}{c}
\begin{bmatrix}
P_{\rm All}\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,SK})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,SK})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,SK})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,SK})_{nm}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{bmatrix}
(Q_{\rm M,FT})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,FT})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,FT})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,FT})_{nm}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{bmatrix}
P_{11}\text{ (frac)} & \dots & P_{1m}\text{ (frac)}\\
\vdots & \ddots & \vdots\\
P_{n1}\text{ (frac)} & \dots & P_{nm}\text{ (frac)}\\
\end{bmatrix}

&

\begin{bmatrix}
P_{1,{\rm All}}\\
\vdots\\
P_{n,{\rm All}}\\
\end{bmatrix}

&

\begin{array}{c}
\begin{bmatrix}
P_{{\rm All}, 1} & \dots & P_{{\rm All}, m}\\
\end{bmatrix}
\\
\\
\\
\end{array}

\end{array}

\end{bmatrix}
</math>


where:
* <math>Q_{\rm M,SK}</math> is mass flow rate of solids to the sinks stream (t/h)
* <math>Q_{\rm M,FT}</math> is mass flow rate of solids to the floats stream (t/h)
* <math>P_{\rm All}</math> is the actual partition of all mass to the sinks stream, computed as <math>\frac{\sum_{i=1}^n\sum_{j=1}^m(Q_{\rm M,SK})_{ij}}{\sum_{i=1}^n\sum_{j=1}^m(Q_{\rm M,F})_{ij}}</math> (frac)
* <math>P_{i,{\rm All}}</math> is the actual partition of all particles of size <math>i</math> to the sinks stream, computed as <math>\frac{\sum_{j=1}^m(Q_{\rm M,SK})_{ij}}{\sum_{j=1}^m(Q_{\rm M,F})_{ij}}</math> (frac)
* <math>P_{{\rm All},i}</math> is the actual partition of all particles of density class <math>j</math> to the sinks stream, computed as <math>\frac{\sum_{i=1}^n(Q_{\rm M,SK})_{ij}}{\sum_{i=1}^n(Q_{\rm M,F})_{ij}}</math> (frac)

=== Example ===

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

{|
|- style="vertical-align:top;"
| [[File:DenseMediumBaguley1.png|left|frame|Figure 3. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]|| [[File:DenseMediumBaguley2.png|left|frame|Figure 4. Example showing the selection of the '''Size''' (red frame), '''Feed''' (green frame) and '''OreSG''' (purple frame) arrays in Excel.]] 
|}

{|
|- style="vertical-align:top;"
| [[File:DenseMediumBaguley3.png|left|frame|Figure 5. Example showing the outline of the '''Results''' (light blue frame) array in Excel.]]
|}

== SysCAD ==

{{SysCAD (Page, Dense Medium, DLL*DenseMedium)}}

==== Drum page ====

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

{{SysCAD (Text, Table Header)}}

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

{{SysCAD (Text, Help Link)}}

|-
! colspan="3" style="text-align:left;" |''Parameters''
|-
|MediumDensity / Rhom
|Input
|Density of the medium.
|-
|MediumViscosity / mu
|Input
|Viscosity of the medium.
|-
|ParticleShapeFactor / k
|Input
|Particle shape factor.
|-
|TerminalVelocity100 / V100
|Input
|Terminal velocity of particles that are 100% recovered to sinks.
|-
|SizeCoefficient / A
|Input
|Coefficient of the size relation.
|-
|SizeConstant / B
|Input
|Constant term of the size relation.

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

|}

{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Partition|ActionL=partition|DestinationU=Sinks|DestinationL=sinks|UnitL=dense medium separator|GravityMetrics=false}}

{{SysCAD (Page, About)}}

==== Additional notes ====

{{SysCAD (Text, No PSD Splits)|gasstream=tail/floats}}

== References ==

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