Shaking Table (Tucker)

Description

This article describes a model for the partition of particles by size and density from a shaking table concentrator. The original model formulation is outlined by Tucker et al. (1991) and subsequent parameter expansions were proposed by Fitzpatrick et al. (2016).^[1]^[2]

Model theory

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Model formulation

Particle ranking

Transition position

Band width

Partition function

Auxiliary model

Limitations

Model parameterisation

Cumulative recovery

Excel

The Tucker shaking table model may be invoked from the Excel formula bar with the following function call:

=mdUnit_ShakingTable_Tucker(Parameters as Range, Size as Range, Density As Range, Feed as Range, Optional AuxParameters 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}{\mathit {CP}}_{k}{\text{ (frac)}}\\a_{1}{\text{ (-)}}\\a_{2}{\text{ (-)}}\\a_{3}{\text{ (-)}}\\b_{0}{\text{ (-)}}\\b_{1}{\text{ (-)}}\\b_{2}{\text{ (-)}}\\b_{3}{\text{ (-)}}\\c_{1}{\text{ (-)}}\\c_{2}{\text{ (-)}}\\c_{3}{\text{ (-)}}\\c_{4}{\text{ (-)}}\\c_{5}{\text{ (-)}}\\{\mathit {SL}}_{\rm {Lower}}{\text{ (mm)}}\\{\mathit {SL}}_{\rm {Upper}}{\text{ (mm)}}\\e_{1}{\text{ (-)}}\\f_{1}{\text{ (-)}}\\f_{2}{\text{ (-)}}\\f_{3}{\text{ (-)}}\\\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}},\;\;\;\;\;\;{\mathit {AuxParameters}}={\begin{bmatrix}(Q_{\rm {M,F}})_{\rm {L}}{\text{ (t/h)}}\\{\mathit {WW}}{\text{ (m}}^{3}{\text{/h)}}\\{\mathit {WW}}_{0}{\text{ (m}}^{3}{\text{/h)}}\\{\mathit {MF}}_{0}{\text{ (t/h)}}\\b_{11}{\text{ (-)}}\\b_{12}{\text{ (-)}}\\b_{13}{\text{ (-)}}\\b_{14}{\text{ (-)}}\\c_{11}{\text{ (-)}}\\c_{12}{\text{ (-)}}\\\end{bmatrix}}^{*}

where:

$d_{i}$ 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)
$(Q_{\rm {M,F}})_{\rm {L}}$ is the mass flow rate of liquids in the feed pulp (t/h)
$^{*}$ indicates an optional array to specify the parameters required by the auxiliary model. If the array is omitted (default), the user-input $b_{1}$ and $c_{1}$ values are used.

Results

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

{\mathit {mdUnit\_ShakingTable\_Tucker}}={\begin{bmatrix}{\begin{array}{c}{\begin{bmatrix}P_{\rm {All}}{\text{ (frac)}}\\\rho _{50}{\text{ (t/m}}^{3}{\text{)}}\\E_{\rm {p}}{\text{ (t/m}}^{3}{\text{)}}\\I{\text{ (-)}}\\b_{1}{\text{ (-)}}\\c_{1}{\text{ (-)}}\\\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,C}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,C}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,C}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,C}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,T}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,T}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,T}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,T}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}T_{11}{\text{ (frac)}}&\dots &T_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\T_{n1}{\text{ (frac)}}&\dots &T_{nm}{\text{ (frac)}}\\\end{bmatrix}}&{\begin{bmatrix}P_{1,{\rm {All}}}{\text{ (frac)}}\\\vdots \\P_{n,{\rm {All}}}{\text{ (frac)}}\\\end{bmatrix}}&{\begin{array}{c}{\begin{bmatrix}P_{{\rm {All}},1}{\text{ (frac)}}&\dots &P_{{\rm {All}},m}{\text{ (frac)}}\\\end{bmatrix}}\\\\\\\end{array}}&{\begin{bmatrix}(\rho _{50})_{1}{\text{ (t/m}}^{3}{\text{)}}\\\vdots \\(\rho _{50})_{n}{\text{ (t/m}}^{3}{\text{)}}\\\end{bmatrix}}&{\begin{bmatrix}(E_{\rm {p}})_{1}{\text{ (t/m}}^{3}{\text{)}}\\\vdots \\(E_{\rm {p}})_{n}{\text{ (t/m}}^{3}{\text{)}}\\\end{bmatrix}}\\\\\\\\\end{array}}\end{bmatrix}}

where:

$Q_{\rm {M,C}}$ is mass flow rate of solids to the concentrate stream (t/h)
$Q_{\rm {M,T}}$ is mass flow rate of solids to the tailing stream (t/h)
$P_{\rm {All}}$ is the actual partition of all mass to the concentrate stream, computed as ${\frac {\sum _{i=1}^{n}\sum _{j=1}^{m}(Q_{\rm {M,C}})_{ij}}{\sum _{i=1}^{n}\sum _{j=1}^{m}(Q_{\rm {M,F}})_{ij}}}$ (frac)
$P_{i,{\rm {All}}}$ is the actual partition of all particles of size $i$ to the concentrate stream, computed as ${\frac {\sum _{j=1}^{m}(Q_{\rm {M,C}})_{ij}}{\sum _{j=1}^{m}(Q_{\rm {M,F}})_{ij}}}$ (frac)
$P_{{\rm {All}},i}$ is the actual partition of all particles of density class $j$ to the concentrate stream, computed as ${\frac {\sum _{i=1}^{n}(Q_{\rm {M,C}})_{ij}}{\sum _{i=1}^{n}(Q_{\rm {M,F}})_{ij}}}$ (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 Density (purple frame) arrays in Excel.

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

SysCAD

The Tucker shaking table model is available from the MetDynamics*GravityConcentrator unit model.

The sections and variable names used in the SysCAD interface are described in detail in the following tables.

Note that a Con and Partition page is provided for each connected concentrate discharge stream of a Gravity Concentrator unit model.

MD_GravityConcentrator 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 MassFracToCon option appears below.
MassFracToCon	Input	Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the concentrate 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.
Options
ShowQFeed	CheckBox	QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed 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.

Con page

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

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Stage
On	Checkbox	This enables the stage. If off, the feed to this stage passes directly to the next stage (or tail) without partition.
Method	Partition (User)	The partition to concentrate for each size interval is defined by the user.
	Partition (Rho50-Ep)	The partition to concentrate for each species and size interval is defined by the Rho50-Ep model.
	Partition (Pivot)	The partition to concentrate for each species and size interval is defined by the Pivot model.
	Partition (Stochastic)	The partition to concentrate for species and each size interval is defined by the Stochastic model.
	Partition (Bazin)	The partition to concentrate for species and each size interval is defined by the Bazin model.
	Jig (King)	The partition to concentrate for species and each size interval is defined by the King jig stratification model.
	Spiral (Tucker)	The partition to concentrate for each species and size interval is defined by the Tucker spiral concentrator model.
	Shaking Table (Tucker)	The partition to concentrate for each species and size interval is defined by the Tucker shaking table model.
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
AuxiliaryModel	CheckBox	Enables the auxiliary model for wash water, mass flow and pulp density parameters.
Parameters
CutterPosition / Cp	Input	Position of the concentrate cutter.
RankSizeExponent / a1	Input	Size exponent of the rank-size equation.
RankDensityCoefficient / a2	Input	Density coefficient of the rank-size equation.
RankDensityExponent / a3	Input	Density exponent of the rank-size equation.
d7Start / b0	Input	d7.5 starting position of the first band.
d75Coefficient / b1	Input	Coefficient of the d7.5 equation.
d75DensityExponent / b2	Input	Density exponent of the d7.5 equation.
d75Exponent / b3	Input	Exponent of the d7.5 equation.
LowerD50Coefficient / c1	Input	Coefficient of the d50 equation for size < SL_Lower.
d50DensityExponent / c2	Input	Density exponent of the d50 equation.
d50Exponent / c3	Input	Exponent of the d50 equation.
MidD50Coefficient / c4	Input	Coefficient of the d50 equation for SL_Lower < size < SL_Upper.
UpperD50Coefficient / c5	Input	Coefficient of the d50 equation for size > SL_Upper.
SL_Upper	Input	Lower size boundary of the d50 equation.
SL_Lower	Input	Upper size boundary of the d50 equation.
InitialPartitionExponent / e1	Input	Exponent of the initial partition equation.
MainPartitionCoefficient / f1	Input	Coefficient of the main partition equation.
MainPartitionD50Coefficient / f2	Input	d50 coefficient of the main partition equation.
MainPartitionExponent / f3	Input	Exponent of the main partition equation.
QvWashWater.Orig / WW0	Input	Original calibration value of the wash water flow rate. Only visible if AuxiliaryModel is selected.
QmSolidsr.Orig / MF0	Input	Original calibration value of the mass flow rate of solids with PSD property in the feed. Only visible if AuxiliaryModel is selected.
WashWaterConstant / b11	Input	Wash water coefficient of the auxiliary model. Only visible if AuxiliaryModel is selected.
PulpDensityExponent / b12	Input	Pulp density exponent of the auxiliary model. Only visible if AuxiliaryModel is selected.
DeltaWashWaterCoeff1 / b13	Input	Delta wash water coefficient of the auxiliary model. Only visible if AuxiliaryModel is selected.
DeltaMassFlowCoeff1 / b14	Input	Delta mass flow rate coefficient of the auxiliary model. Only visible if AuxiliaryModel is selected.
DeltaWashWaterCoeff2 / c11	Input	Delta wash water coefficient of the auxiliary model. Only visible if AuxiliaryModel is selected.
DeltaMassFlowCoeff2 / c12	Input	Delta mass flow rate coefficient of the auxiliary model. Only visible if AuxiliaryModel is selected.
Liquids
LiquidsSeparMethod	Split To Con (User)	Liquids are split to concentrate by a user-defined fraction of liquids in the feed.
	Con Solids Fraction	Sufficient liquids mass is recovered to the concentrate stream to yield the user-defined concentrate solids mass fraction value (if possible).
	Con Liquids Fraction	Sufficient liquids mass is recovered to the concentrate stream to yield the user-defined concentrate liquids mass fraction value (if possible).
ConSolidsFracReqd / Con.SfReqd	Input	Required value of the mass fraction of solids in the concentrate stream. Only visible if Con Solids Fraction is selected.
ConLiquidsFracReqd / Con.LfReqd	Input	Required value of the mass fraction of liquids in the concentrate stream. Only visible if Con Liquids Fraction is selected.
LiqSplitToCon / Con.LiqSplit	Input/Display	The fraction of feed liquids recovered to the concentrate stream.
Results
QvWashWater.Sim / WW	Display	Simulated wash water flow rate.
QmSolids.Sim / MF	Display	Simulated mass flow rate of solids with PSD property in feed.
CutDensity / Rho50	Display	Unsized cut density (Rho50).
EcartProbable / Ep	Display	Unsized Ecart Probable (Ep).
Imperfection / I	Display	Unsized partition Imperfection (I).

Partition page

The Partition page is used to display (or specify) the partition by species/component/element/individual phase 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.
CumulativePartition
Method	Model / User	Select model-calculated or user-defined cumulative 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.
CutDensity / Rho50	Display	Cut density (Rho50) of all particles in each size interval.
EcartProbable / Ep	Display	Ecart Probable (Ep) of all particles in each size interval.
All (All column)	Display	Actual overall cumulative partition to concentrate of all solid species, for each size interval. Excludes solid species not present in the gravity concentrator feed.
CumulativePartition	Display	Cumulative partition to concentrate 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 gravity concentrator feed or not.
All (All row, All column)	Display	Displays the actual, total, cumulative partition of all solids with a particle size distribution property in the feed to concentrate. Excludes solid species not present in the gravity concentrator feed.
All (All row, per species)	Display	Actual overall cumulative partition to concentrate for each solid species, for all size intervals in that species. Excludes solid species not present in the gravity concentrator feed.
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.
CutDensity / Rho50	Display	Cut density (Rho50) of all particles in each size interval.
EcartProbable / Ep	Display	Ecart Probable (Ep) of all particles in each size interval.
All (All column)	Display	Actual overall partition to concentrate of all solid species, for each size interval. Excludes solid species not present in the gravity concentrator feed.
Partition	Display	Partition to concentrate 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 gravity concentrator 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 concentrate. Excludes solid species not present in the gravity concentrator feed.
All (All row, per species)	Display	Actual overall partition to concentrate for each solid species, for all size intervals in that species. Excludes solid species not present in the gravity concentrator feed.
CmpPartition
Components		Hides or shows component partition table.
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 concentrate of all solid components, for each size interval. Excludes solid components not present in the gravity concentrator feed.
CmpPartition	Display	Partition to concentrate for each size interval, in each solid component, as determined by the selected model or user defined value. Excludes solid components not present in the gravity concentrator feed.
All (All row, All column)	Display	Displays the actual, total, partition of all solid components with a particle size distribution property in the feed to concentrate. Excludes solid components not present in the gravity concentrator feed.
All (All row, per component)	Display	Actual overall partition to concentrate for each solid component, for all size intervals in that component. Excludes solid components not present in the gravity concentrator feed.
ElePartition
Elements		Hides or shows element partition table.
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 concentrate of all solid elements, for each size interval. Excludes solid elements not present in the gravity concentrator feed.
ElePartition	Display	Partition to concentrate for each size interval, in each solid element, as determined by the selected model or user defined value. Excludes solid elements not present in the gravity concentrator feed.
All (All row, All column)	Display	Displays the actual, total, partition of all solid elements with a particle size distribution property in the feed to concentrate. Excludes solid elements not present in the gravity concentrator feed.
All (All row, per element)	Display	Actual overall partition to concentrate for each solid element, for all size intervals in that component. Excludes solid elements not present in the gravity concentrator feed.
IPhPartition
IPhases		Hides or shows individual phases partition table.
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 concentrate of all solid individual phases, for each size interval. Excludes solid individual phases not present in the gravity concentrator feed.
IPhPartition	Display	Partition to concentrate for each size interval, in each solid individual phase, as determined by the selected model or user defined value. Excludes solid individual phases not present in the gravity concentrator feed.
All (All row, All column)	Display	Displays the actual, total, partition of all solid individual phases with a particle size distribution property in the feed to concentrate. Excludes solid individual phases not present in the gravity concentrator feed.
All (All row, per individual phase)	Display	Actual overall partition to concentrate for each solid individual phase, for all size intervals in that component. Excludes solid individual phases not present in the gravity concentrator 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 stream without split.

External links

Met Dynamics - Gravity Concentrator (help.syscad.net)

References

↑ Tucker, P., Lewis, K.A. and Wood, P., 1991. Computer optimisation of a shaking table. Minerals Engineering, 4(3-4), pp.355-367.
↑ Fitzpatrick, R.S., Ghorbani, Y., Hegarty, P. and Rollinson, G., 2016. Quantitative mineralogy for improved modelling of shaking tables. In XXVIII International Mineral Processing Congress. Canadian Institute of Mining, Metallurgy and Petroleum.

[Tucker_et_al._(1991)-1] Tucker, P., Lewis, K.A. and Wood, P., 1991. Computer optimisation of a shaking table. Minerals Engineering, 4(3-4), pp.355-367.

[Fitzpatrick_et_al._(2016)-2] Fitzpatrick, R.S., Ghorbani, Y., Hegarty, P. and Rollinson, G., 2016. Quantitative mineralogy for improved modelling of shaking tables. In XXVIII International Mineral Processing Congress. Canadian Institute of Mining, Metallurgy and Petroleum.

[1]

[2]

Shaking Table (Tucker)

Contents

Description

Model theory

Model formulation

Particle ranking

Transition position

Band width

Partition function

Auxiliary model

Limitations

Model parameterisation

Cumulative recovery

Excel

Inputs

Results

Example

SysCAD

MD_GravityConcentrator page

Con page

Partition page

About page

Additional notes

See also

External links

References

Navigation menu

Shaking Table (Tucker)

Description

Model theory

Model formulation

Particle ranking

Transition position

Band width

Partition function

Auxiliary model

Limitations

Model parameterisation

Cumulative recovery

Excel

Inputs

Results

Example

SysCAD

MD_GravityConcentrator page

Con page

Partition page

About page

Additional notes

See also

External links

References

Navigation menu

Search