Falcon Concentrator (Kroll-Rabotin)
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Description
This article describes the Kroll-Rabotin et al. (2013) model for the partition of particles by size and density from a smooth bowl Falcon centrifugal concentrator.[1]
The Falcon concentrator is a centrifugal gravity separator used for recovering fine and ultrafine high-density minerals. This model applies to smooth-bowl units, such as the UF and C series.
In these units, slurry forms a thin flowing film along the inner wall of a rapidly rotating bowl. Particles are transported within this film under a strong centrifugal field. Separation occurs as particles move radially at different rates according to their size and density. Denser or coarser particles reach the wall and are retained, while lighter or finer particles remain in the flow and report to the overflow.
This behaviour is governed by the balance between centrifugal acceleration and viscous drag acting on each particle. It is influenced by particle properties, slurry concentration, bowl geometry, and rotation speed.
Falcon SB series concentrators operate on a different principle using fluidised retention zones and are not described by this model.
Model theory
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Other properties
Limitations
Excel
The Falcon concentrator model may be invoked from the Excel formula bar with the following function call:
=mdUnit_FalconConcentrator_KR(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 ([math]\displaystyle{ i }[/math]) x column ([math]\displaystyle{ j }[/math]) format:
- [math]\displaystyle{ \mathit{Parameters} = \begin{bmatrix} R_{\rm max}\text{ (m)}\\ R_{\rm min}\text{ (m)}\\ H_{\rm bowl}\text{ (m)}\\ \omega\text{ (rpm)}\\ \lambda_0\text{ (-)}\\ \lambda_{\phi}\text{ (-)}\\ \mu\text{ (Pa.s)}\\ \rho_{\rm L}\text{ (t/m}^3\text{)}\\ (Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\ \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]\displaystyle{ d_i }[/math] is the size of the square mesh interval that feed mass is retained on (mm)
- [math]\displaystyle{ Q_{\rm M,F} }[/math] is feed solids mass flow rate by size and ore type (t/h)
- [math]\displaystyle{ (Q_{\rm M,F})_{\rm L} }[/math] is the mass flow rate of liquids in the feed (t/h)
- [math]\displaystyle{ \rho_{\rm S} }[/math] is the density of solids (t/m3)
- [math]\displaystyle{ \rho_{\rm L} }[/math] is the density of liquids (t/m3)
Results
The results are displayed in Excel as an array corresponding to the matrix notation below:
- [math]\displaystyle{ \mathit{mdUnit\_FalconConcentrator\_KR} = \begin{bmatrix} \begin{array}{c} \begin{bmatrix} \mathrm{Iterations}\\ Q\text{ (m}^3\text{/s)}\\ \phi\text{ (v/v)}\\ \mathrm{RCF}\text{ (-)}\\ h\text{ (m)}\\ t_{\rm Res}\text{ (s)}\\ P_{\rm All}\text{ (frac)}\\ \rho_{50}\text{ (t/m}^3\text{)}\\ E_{\rm p}\text{ (t/m}^3\text{)}\\ I\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} C_{11}\text{ (frac)} & \dots & C_{1m}\text{ (frac)}\\ \vdots & \ddots & \vdots\\ C_{n1}\text{ (frac)} & \dots & C_{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} }[/math]
where:
- [math]\displaystyle{ \mathrm{Iterations} }[/math] is the number of iterations required to converge the [math]\displaystyle{ C - \rho_{\rm s} }[/math] dependence
- [math]\displaystyle{ Q_{\rm M,C} }[/math] is mass flow rate of solids to the concentrate stream (t/h)
- [math]\displaystyle{ Q_{\rm M,T} }[/math] is mass flow rate of solids to the tailing stream (t/h)
- [math]\displaystyle{ P_{\rm All} }[/math] is the actual partition of all mass to the concentrate stream, computed as [math]\displaystyle{ \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}} }[/math] (frac)
- [math]\displaystyle{ P_{i,{\rm All}} }[/math] is the actual partition of all particles of size [math]\displaystyle{ i }[/math] to the concentrate stream, computed as [math]\displaystyle{ \frac{\sum_{j=1}^m(Q_{\rm M,C})_{ij}}{\sum_{j=1}^m(Q_{\rm M,F})_{ij}} }[/math] (frac)
- [math]\displaystyle{ P_{{\rm All},i} }[/math] is the actual partition of all particles of density class [math]\displaystyle{ j }[/math] to the concentrate stream, computed as [math]\displaystyle{ \frac{\sum_{i=1}^n(Q_{\rm M,C})_{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.
 Figure 1. Example showing the selection of the Parameters (blue frame) array in Excel. |
 Figure 2. Example showing the selection of the Size (red frame), Feed (green frame) and Density (purple frame) arrays in Excel. |
 Figure 3. Example showing the outline of the Results (light blue frame) array in Excel. |
SysCAD
The Falcon concentrator 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:
|
| 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. | |
| Falcon (Kroll-Rabotin) | The partition to concentrate for each species and size interval is defined by the Kroll-Rabotin Falcon concentrator model. | |
| Knelson (Chen) | The partition to concentrate for each species and size interval is defined by the Chen Knelson concentrator model. | |
| HelpLink | 
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Opens a link to this page using the system default web browser. Note: Internet access is required. |
| Parameters | ||
| OuterBowlRadius / Rmax | Input | Maximum radial length of the bowl. |
| InnerBowlRadius / Rmin | Input | Minimum radial length of the bowl. |
| BowlHieght / Hbowl | Input | Height of the bowl. |
| RotationRate / omega | Input | Rotational speed of the bowl. |
| CalibrationCoeff / lambda0 | Input | Calibration coefficient of the partition equation. |
| HinderedSetCoeff / lambdaPhi | Input | Hindered settling coefficient of the partition equation. |
| 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 | ||
| Feed.VolFlow / Q | Display | Volumetric flow rate of the feed. |
| Feed.VolFrac / phi | Display | Volumetric fraction of solids in the feed. |
| Feed.LiqViscosity / mu | Display | Viscosity of liquids in the feed. |
| SuspensionDensity / Rhos | Display | Density of the suspension in the bowl. |
| GForce | Display | Gravitational (G) force developed by the rotating bowl. |
| FilmThickness / h | Display | Average thickness of the flowing film inside the bowl. |
| ResidenceTime / tRes | Display | Average residence time of overflowing particles in the film. |
| 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. |
| 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 |
|
| Partition | Display |
|
| All (All row, All column) | Display |
|
| All (All row, per species) | Display |
|
| 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 |
|
| CmpPartition | Display |
|
| All (All row, All column) | Display |
|
| All (All row, per component) | Display |
|
| 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 |
|
| ElePartition | Display |
|
| All (All row, All column) | Display |
|
| All (All row, per element) | Display |
|
| 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 |
|
| IPhPartition | Display |
|
| All (All row, All column) | Display |
|
| All (All row, per individual phase) | Display |
|
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
References
- ↑ Kroll-Rabotin, J.S., Bourgeois, F. and Climent, É., 2013. Physical analysis and modeling of the Falcon concentrator for beneficiation of ultrafine particles. International Journal of Mineral Processing, 121, pp.39-50.