Dense Medium Drum (Baguley)

Description

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

Model theory

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.). Chart series represent density classes.

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.). 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.

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

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

where particle size effects are incorporated via the $x$ term:

$\ln(x)={\dfrac {A}{{{\bar {d}}_{i}}^{2}}}+B$

and:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,q\}$, $q$ is the number of size intervals
• $j$ is the index of the density class, $j=\{1,2,\dots ,m\}$, $m$ is the number of density classes
• $V_{100}$ is the terminal velocity of particles of which 100% report to sinks (m/s)
• $V_{\rm {t}}$ is the terminal velocity of particles (m/s)
• $A$ is a coefficient of the size expression (-)
• $B$ is constant of the size expression (-)
• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)

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

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

$V_{\rm {t}}^{*}={\dfrac {{\delta _{0}}^{2}}{4d^{*}}}\left[\left(1+{\dfrac {4{d^{*}}^{1.5}}{{C_{0}}^{0.5}{\delta _{0}}^{2}}}\right)^{0.5}-1\right]^{2}$

where $d^{*}$ is the dimensionless particle size computed from:

$d^{*}={\dfrac {\bar {d}}{p}}$

and ${\bar {d}}$ is the geometric mean size of particles in a size interval (with units of m in the $d^{*}$ equation).

The particle-fluid characteristic size parameter, $p$, is:

$p=\left({\dfrac {3\mu ^{2}}{4(\Delta \rho )\rho _{\rm {m}}g}}\right)^{\frac {1}{3}}$

where:

• $\mu$ is the apparent equivalent Newtonian medium viscosity (Pa.s)
• $\Delta \rho$ is density difference between a particle and the medium (kg/m³)
• $\rho _{\rm {m}}$ is medium density (kg/m³)
• $g$ is acceleration due to gravity (m/s²)

The shape-dependent parameters $C_{0}$ and $\delta _{0}$ are calculated from:

$\ln C_{0}=-5.99k+1.95$

$\delta _{0}=9.97k+2.94$

where $k$ is a particle shape coefficient, assumed to be 0.225 for spheres.

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

$V_{\rm {t}}=V_{\rm {t}}^{*}q$

where the particle-fluid characteristic velocity parameter, $q$, is:

$q=\left({\dfrac {4(\Delta \rho )\mu g}{3{\rho _{\rm {m}}}^{2}}}\right)^{\frac {1}{3}}$

Excel

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

=mdUnit_DMDrum_Baguley(Parameters as Range, Size as Range, Density 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:

${\mathit {Parameters}}={\begin{bmatrix}\rho _{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\mu {\text{ (mPa.s)}}\\k\\V_{100}{\text{ (m/s)}}\\A\\B\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {Size}}={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {Feed}}={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {Density}}={\begin{bmatrix}(\rho _{\rm {s}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {s}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}}$

where:

• $n$ is the number of size intervals
• $m$ is the number of density classes
• $d_{i}$ is the size of the square mesh interval that feed mass is retained on (mm)
• $Q_{\rm {M,F}}$ is feed solids mass flow rate by size and ore type (t/h)
• $\rho _{\rm {S}}$ is the particle density (t/m³)

Results

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

${\mathit {mdUnit\_DenseMedium\_Baguley}}={\begin{bmatrix}{\begin{array}{c}{\begin{bmatrix}P_{\rm {All}}{\text{ (frac)}}\\\end{bmatrix}}\\\\\\\end{array}}{\begin{array}{cccccc}{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,SK}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,SK}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,SK}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,SK}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,FT}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,FT}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,FT}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,FT}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}P_{11}{\text{ (frac)}}&\dots &P_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\P_{n1}{\text{ (frac)}}&\dots &P_{nm}{\text{ (frac)}}\\\end{bmatrix}}&{\begin{bmatrix}P_{1,{\rm {All}}}\\\vdots \\P_{n,{\rm {All}}}\\\end{bmatrix}}&{\begin{array}{c}{\begin{bmatrix}P_{{\rm {All}},1}&\dots &P_{{\rm {All}},m}\\\end{bmatrix}}\\\\\\\end{array}}\end{array}}\end{bmatrix}}$

where:

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

Example

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

Figure 3. Example showing the selection of the Parameters (blue frame) array in Excel.

Figure 4. Example showing the selection of the Size (red frame), Feed (green frame) and OreSG (purple frame) arrays in Excel.

Figure 5. Example showing the outline of the Results (light blue frame) array in Excel.

SysCAD

MD_DenseMedium 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 MassFracToSinks option appears below.
MassFracToSinks	Input	Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the sinks 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.
Method	Partition (User)	The partition to sinks for each size interval is defined by the user.
	Partition (Rho50-Ep)	The partition to sinks for each size interval is defined by the Rh050-Ep model.
	Partition (Pivot)	The partition to sinks for each size interval is defined by the Pivot model.
	Partition (Stochastic)	The partition to sinks for each size interval is defined by the Stochastic model.
	Drum (Baguley)	The Baguley dense medium drum model is used to determine the partition of solids to sinks and floats 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.
ShowQSinks	CheckBox	QSinks and associated tab pages (eg Sp) will become visible, showing the properties of the sinks stream.
ShowQFloats	CheckBox	QFloats and associated tab pages (eg Sp) will become visible, showing the properties of the floats 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.

Drum page

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

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
DMDrum
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
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.
Liquids
LiquidsSeparMethod	Split To Sinks (User)	Liquids are split to sinks by a user-defined fraction of liquids in the feed.
	Sinks Solids Fraction	Sufficient liquids mass is recovered to the sinks stream to yield the user-defined sinks solids mass fraction value (if possible).
	Sinks Liquids Fraction	Sufficient liquids mass is recovered to the sinks stream to yield the user-defined sinks liquids mass fraction value (if possible).
SinksSolidsFracReqd / Sinks.SfReqd	Input	Required value of the mass fraction of solids in the sinks stream. Only visible if Sinks Solids Fraction is selected.
SinksLiquidsFracReqd / Sinks.LfReqd	Input	Required value of the mass fraction of liquids in the sinks stream. Only visible if Sinks Liquids Fraction is selected.
LiqSplitToSinks / Sinks.LiqSplit	Input/Display	The fraction of feed liquids recovered to the sinks 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 sinks of all solid species, for each size interval. Excludes solid species not present in the dense medium separator feed.
Partition	Display	Partition to sinks 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 dense medium separator 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 sinks. Excludes solid species not present in the dense medium separator feed.
All (All row, per species)	Display	Actual overall partition to sinks for each solid species, for all size intervals in that species. Excludes solid species not present in the dense medium separator 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 tail/floats stream without split.