Partition (Size, Whiten-Beta)

Description

This article describes Whiten's expression for the efficiency of recovery to overflow, modified to represent observed 'fish hook' behaviour in the finer sizes.^[1]

Model theory

Figure 1. Whiten-Beta partitions to overflow, with (orange line) and without (blue line) use of the

\beta

parameter.

The Whiten-Beta expression for partition to overflow is:

E_{\rm {oa}}({\bar {d}}_{i})=C\left[{\dfrac {\left(1+\beta \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)(\exp(\alpha )-1)}{\exp \left(\alpha \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)+\exp(\alpha )-2}}\right]

where:

$i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$ , $n$ is the number of size intervals
$E_{\rm {oa}}({\bar {d}}_{i})$ is the fraction of particles of size interval $i$ in the feed reporting to the overflow stream (frac)
${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)
$d_{\rm {50c}}$ is the corrected size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
$C$ is the fraction of feed liquids (or fines) split to overflow (frac)
$\alpha$ is a parameter representing the sharpness of separation
$\beta$ is a term introduced to accommodate the so-called fish-hook effect, and controls the initial rise in the efficiency curve at finer sizes
$\beta ^{*}$ is computed to ensure the Whiten-Beta function preserves the definition of $d_{\rm {50c}}$ in the presence of the fish-hook, i.e. $E=0.5C$ at $d_{\rm {50c}}$

The value of $\beta ^{*}$ is not an input parameter and is computed numerically, by iteration, since by rearrangement:

\beta ^{*}={\dfrac {\ln {\big [}\exp(\alpha )+2\beta \beta ^{*}(\exp(\alpha )-1){\big ]}}{\alpha }}

When the value of $\beta$ is zero, the Whiten-Beta equation reverts to:

E_{{\rm {oa}}i}=C\left[{\dfrac {\exp(\alpha )-1}{\exp \left(\alpha {\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)+\exp(\alpha )-2}}\right]

Excel

The Whiten-Beta efficiency curve model may be invoked from the Excel formula bar with the following function call:

=mdPartition_WhitenBeta(MeanSize as Range, alpha as Double, d50c as Double, C as double, Optional beta as Double = 0, Optional returnBetaStar as Boolean = false)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

The input parameters and model results are defined below in matrix notation, along with an example image showing the selection of the same cells in the Excel interface:

${\begin{aligned}MeanSize&={\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\\\end{bmatrix}}\\\\alpha&={\big [}\alpha {\big ]}\\d50c&={\big [}d_{50{\rm {c}}}{\text{ (mm)}}{\big ]}\\C&={\big [}C{\text{ (frac)}}{\big ]}\\Beta&={\big [}\beta {\big ]}\\returnBetaStar&={\big [}({\text{True / False)}}{\big ]}\\\end{aligned}}$	$mdPartition\_WhitenBeta={\begin{bmatrix}(E_{\rm {oa}})_{1}{\text{ (frac)}}&\beta ^{*}\\\vdots &[-]\\(E_{\rm {oa}})_{n}{\text{ (frac}}&[-]\\\end{bmatrix}}$	Figure 2. Example showing the selection of the input parameters (shaded cells), and Results (light blue frame) array in Excel. The parameter $returnBetaStar={\text{False}}$ in this example.
where $\beta ^{*}$ is omitted from the results array if $returnBetaStar={\text{False}}$ .

SysCAD

The Whiten-Beta partition is available from the MetDynamics*Hydrocyclone unit model.

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

MD_Hydrocyclone 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 MassFracToUF option appears below.
MassFracToUF	Input	Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the underflow stream when the model is off.
Method	Partition (User)	The partition to overflow for each size interval is defined by the user. Different values can be used for different solids.
	Partition (Reid-Plitt)	The partition to overflow for each size interval is defined by a Reid-Plitt efficiency curve. Different parameters can be used for different solids.
	Partition (Whiten-Beta)	The partition to overflow for each size interval is defined by a Whiten-Beta efficiency curve. Different parameters can be used for different solids.
	Nageswararao	The Nageswararao model is used to determine the partition of solids to underflow and overflow for each size interval.
	Narasimha-Mainza (2014)	The Narasimha-Mainza (2014) model is used to determine the partition of solids to underflow and overflow for each size interval.
	Narasimha-Mainza (Multi)	The Narasimha-Mainza (Multi) model is used to determine the partition of solids to underflow and overflow for each size interval.
	Plitt	The Plitt model is used to determine the partition of solids to underflow and overflow for each size interval.
RopingCalcs	CheckBox	Show addition calculations that predict the onset of cyclone underflow roping.
Options
ShowQFeed	CheckBox	QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
ShowQOF	CheckBox	QOF and associated tab pages (eg Sp) will become visible, showing the properties of the overflow stream.
ShowQUF	CheckBox	QUF and associated tab pages (eg Sp) will become visible, showing the properties of the underflow 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.

Cyclone page

The Cyclone page is used to specify the input parameters for the hydrocyclone model.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
WhitenBeta
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
OreSpecific	CheckBox	Ore-specific partitions, allows the Whiten-Beta equation parameters to be separately input for all species. Default is all species have the same set of single input properties. This option is only available if there is more than one species in the project with the size distribution property.
Alpha	Input	The Whiten-Beta equation alpha parameter.
d50c	Input	The Whiten-Beta equation d50c corrected cut size parameter.
C	Input	The Whiten-Beta equation C parameter.
Beta	Input	The Whiten-Beta equation beta parameter.
Beta*	Display	The Whiten-Beta equation Beta* parameter corresponding to the selected value of Beta.
Liquids
LiquidsSeparMethod	Split To UF (User)	Liquids are split to underflow by a user-defined fraction of liquids in the feed.
	Split To UF (1-C)	Liquids are split to underflow by the C (overflow) fraction specified for the first species in the model.
	UF Solids Fraction	Sufficient liquids mass is recovered to the underflow stream to yield the user-defined underflow solids mass fraction value (if possible).
	UF Liquids Fraction	Sufficient liquids mass is recovered to the underflow stream to yield the user-defined underflow liquids mass fraction value (if possible).
UFSolidsFracReqd / UF.SfReqd	Input	Required value of the mass fraction of solids in the underflow stream. Only visible if UF Solids Fraction is selected.
UFLiquidsFracReqd / UF.LfReqd	Input	Required value of the mass fraction of liquids in the underflow stream. Only visible if UF Liquids Fraction is selected.
LiqSplitToUF / UF.LiqSplit	Input/Display	The fraction of liquids recovered to underflow.

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 overflow of all solid species, for each size interval. Excludes solid species not present in the cyclone feed.
Partition	Display	Partition to overflow 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 cyclone 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 overflow. Excludes solid species not present in the cyclone feed.
All (All row, per species)	Display	Actual overall partition to overflow for each solid species, for all size intervals in that species. Excludes solid species not present in the cyclone feed.

Roping page

This page displays the results for hydrocyclone roping limit calculations. The page is only visible if Roping is selected on the MD_Hydrocyclone page.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Roping
Underflow
SolidsVolFrac / Svf	Display	Volume fraction of solids in the cyclone underflow stream.
SPOC	Display	SPOC volume fraction of solids in the cyclone underflow roping limit.

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.

References

↑ Napier-Munn, T.J., Morrell, S., Morrison, R.D. and Kojovic, T., 1996. Mineral comminution circuits: their operation and optimisation. Julius Kruttschnitt Mineral Research Centre, Indooroopilly, QLD.

[Napier-Munn_et_al._(1996)-1] Napier-Munn, T.J., Morrell, S., Morrison, R.D. and Kojovic, T., 1996. Mineral comminution circuits: their operation and optimisation. Julius Kruttschnitt Mineral Research Centre, Indooroopilly, QLD.

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Partition (Size, Whiten-Beta)

Contents

Description

Model theory

Excel