Partition (Density, Stochastic): Difference between revisions

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{{Excel (Text, Help, No Arguments)}}
{{Excel (Text, Help, No Arguments)}}
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:
:{|
|- style="vertical-align:top;"
|
<math>
\begin{align}
MeanSize & =
\begin{bmatrix}
\bar{d}_1\text{ (mm)}\\
\vdots\\
\bar{d}_n\text{ (mm)}\\
\end{bmatrix}\\
\\
RD & = \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}\\
\\
A & = \big [A \big ]\\
c & = \big [c \big ]\\
B & = \big [B \big ]\\
rhop & = \big [\rho_{\rm p} \text{ (t/m}^3\text{))}\big ]\\
hf & = \big [h_{\rm f}\text{ (m/m)} \big ]\\
D & = \big [D \big ]\\
hl & = \big [h_{\rm l}\text{ (m/m)} \big ]\\
t & = \big [t\text{ (s)} \big ]\\
returnPartitionMetrics & = \big [(\text{True / False)} \big ]\\
\\
\\
mdPartition\_StochasticRao & =
\begin{bmatrix}
\begin{bmatrix}
Y_{11}\text{ (t/h)} & \dots & Y_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
Y_{n1}\text{ (t/h)} & \dots & Y_{nm}\text{ (t/h)}\\
\end{bmatrix}
&
\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}
&
\begin{array}{c}
\begin{bmatrix}
(d_{50})_{1}\text{ (mm)} & \dots & (d_{50})_m\text{ (mm)}\\
\end{bmatrix}^{\rm *}
\\
\\
\\
\\
\end{array}
\end{bmatrix}
\end{align}</math>
where:
* <math>n</math> is the number of size intervals
* <math>*</math> indicates optional results returned if <math>returnPartitionMetrics = True</math>
|- style="vertical-align:top;"
| [[File:PartitionStochastic1.png|frame|Figure 2. Example showing the selection of the input parameters (purple, green, pink, brown, teal, blue, red and purple cells), and the '''MeanSize''' (blue frame), '''RD''' (red frame) and '''Results''' (light blue frame) arrays in Excel. The parameter <math>returnPartitionMetrics =\text{True}</math> in this example.]]
|}


== SysCAD ==
== SysCAD ==


{{Under construction|section}}
The Stochastic partition is available from the '''{{Name (Text, Company Name)|nospace=1}}*GravityConcentrator''' unit model.
 
The sections and variable names used in the SysCAD interface are described in detail in the following tables.
 
{{SysCAD (Page, Gravity Concentrator, DLL*GravityConcentrator)}}
 
==== Stochastic page ====
 
The Stochastic page is used to specify the input parameters for the Stochastic model.
 
{{SysCAD (Text, Table Header)}}
 
|-
! colspan="3" style="text-align:left;" |''Stochastic''
 
{{SysCAD (Text, Help Link)}}
 
|-
! colspan="3" style="text-align:left;" |''Parameters''
|-
|A
|Input
|Partition surface parameter, gravitation and viscous forces.
|-
|c
|Input
|Partition surface parameter, turbulence.
|-
|B
|Input
|Partition surface parameter, fluid drift force.
|-
|PivotDensity / Rhop
|Input
|Density at pivot point.
|-
|hf
|Input
|Feed particle input bed position.
|-
|D
|Input
|Partition surface parameter, standard deviation of particle velocities.
|-
|hf
|Input
|Slice position of the particle bed.
|-
|t
|Input
|Separation period of particles.
 
{{SysCAD (Text, Gravity Concentrator, Liquids)|method=1}}
 
|}
 
{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Partition|ActionL=partition|DestinationU=Overflow|DestinationL=overflow|UnitL=cyclone|GravityMetrics=true}}
 
{{SysCAD (Page, About)}}


== References ==
== References ==

Revision as of 06:24, 14 May 2023

Description

This article describes Rao's Stochastic equation for the partition of particles by size and density in gravity separators.[1][2]

Model theory

Under construction icon-blue.svg.png This section is currently under construction. Please check back later for updates and revisions.

Excel

Under construction icon-blue.svg.png This section is currently under construction. Please check back later for updates and revisions.

The Stochastic partition equation may be invoked from the Excel formula bar with the following function call:

=mdPartition_StochasticRao(MeanSize as Range, RD as Range, A as Double, c as Double, B as Double, rhop as Double, hf as Double, D as double, hl as Double, t as Double, Optional returnPartitionMetrics as Bool = 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:


where:

  • is the number of size intervals
  • indicates optional results returned if


Figure 2. Example showing the selection of the input parameters (purple, green, pink, brown, teal, blue, red and purple cells), and the MeanSize (blue frame), RD (red frame) and Results (light blue frame) arrays in Excel. The parameter in this example.

SysCAD

The Stochastic partition 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.

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.
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.

Stochastic page

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

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Stochastic
HelpLink ButtonModelHelp.png Opens a link to this page using the system default web browser. Note: Internet access is required.
Parameters
A Input Partition surface parameter, gravitation and viscous forces.
c Input Partition surface parameter, turbulence.
B Input Partition surface parameter, fluid drift force.
PivotDensity / Rhop Input Density at pivot point.
hf Input Feed particle input bed position.
D Input Partition surface parameter, standard deviation of particle velocities.
hf Input Slice position of the particle bed.
t Input Separation period of particles.
Liquids
LiquidsSeparMethod Split To Con (User) Liquids are split to concentrate by a user-defined fraction of liquids in the feed.
Split To Con (Model) Liquids are split to concentrate by a fraction determined by the selected model.
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.

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.

PartitionMetrics
d50 Display Cut size (d50) of separation of each ore 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 of all particles in each size interval.
PivotPartition / Yp Display Partition at pivot point.

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 ButtonLicensingHelp.png Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information ButtonCopyToClipboard.png 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 ButtonBrowse.png 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

  1. Rao, B.V., Kapur, P.C. and Konnur, R., 2003. Modeling the size–density partition surface of dense-medium separators. International Journal of Mineral Processing, 72(1-4), pp.443-453.
  2. Rao, B.V., 2021. An improved stochastic model to describe partition surfaces of entire segregated batch jig bed. Minerals Engineering, 170, p.107064.