AG/SAG Mill (Variable Rates)

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Description

This article describes an implementation of the Autogenous (AG) and Semi-Autogenous (SAG) mill model originated by Leung (1987) and extended with variable breakage rates by Morrell and Morrison (1996).[1][2][3]

The formulation is referred to in the associated literature as the "Variable Rates" model (Morrell et al., 2001).[4]

Model theory

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Excel

The Variable Rates AG/SAG mill model may be invoked from the Excel formula bar with the following function call:

=mdUnit_AGSAG_VariableRates(Parameters as Range, Size as Range, MillNewFeed as Range, OreSG as Range, BallSizing as Range, RConst as Range, OreBreakageParams as Range, Optional MillRecycleFeed as Range = Nothing)

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 () x column () format:


where:

  • is the mass flow feed rate of liquids into the mill (t/h)
  • is the Specific Gravity or density of liquids in the feed (- or t/m3)
  • is an index of the Appearance function to view in the results
  • is an index of the Appearance function to view in the results
  • is the number of ore types
  • is the number of intervals of the external mesh series
  • is the number of intervals of the ball mesh series below the top size, including the submesh
  • is the size of the external square mesh interval that feed mass is retained on (mm)
  • , i.e. descending size order from top size () to sub mesh ()
  • is the size of the square mesh interval that balls are retained on (mm)
  • is the mass fraction of balls retained on ball mesh series interval (% w/w)
  • indicates the array is an optional input parameter, and is set to null if omitted

Results

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

where:

  • is the number of internal computation steps required to converge the load
  • is the numerical error of the converged load approximation
  • is the flow rate of pulp into the mill (m3/h)
  • is the rotational rate of the mill (rpm)
  • is the mass of ore solids in the mill (t)
  • is the mass of liquids in the mill (t)
  • is the mass of balls in the mill (t)
  • is the total mass of ore, liquids and balls in the mill (t)
  • is product mass flow rate (t/h)
  • is the geometric mean size of the internal mesh series interval that mass is retained on (mm)

Example

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

Figure 8. Example showing the selection of the Parameters (blue frame) array in Excel.
Figure 9. Example showing the selection of the Size (dark red frame), OreSG (green frame), MillNewFeed (purple frame) and MillRecycleFeed (light red frame) arrays in Excel.
Figure 10. Example showing the selection of the BallSizing (purple frame), RConst (brown frame), and OreBreakageParams (teal frame) arrays in Excel.
Figure 11. Example showing the outline of the Results (light blue frame) array in Excel.

SysCAD

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

MD_Mill 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 material will pass straight through the mill with no change to the size distribution.
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 product is multiplied back by the same value and returned to the SysCAD product stream.
  • All unit model result values are shown per parallel unit.
Method Fixed Discharge The discharge particle size distribution is user defined. Different distributions can be used for different solids.
AG/SAG (Variable Rates) The Variable Rates AG/SAG mill model is used to determine the mill product size distribution. Different parameters can be used for different solids.
Rod Mill (Lynch) The Lynch rod mill model is used to determine the mill product size distribution. Different parameters can be used for different solids.
Ball (Perfect Mixing) The Perfect Mixing ball mill model (steady-state or dynamic) is used to determine the mill product size distribution. Different parameters can be used for different solids.
Stirred (Perfect Mixing) The Perfect Mixing stirred mill model (steady-state or dynamic) is used to determine the mill product size distribution. Different parameters can be used for different solids.
Mill (Herbst-Fuerstenau) The Herbst-Fuerstenau model is used to determine the mill product size distribution. Different parameters can be used for different solids.
PowerModels CheckBox Show alternative mill power model calculations on the Power page.
MediaTrajectory CheckBox Show mill media rolling, sliding and free flight trajectory computations on the MediaTraj page.
MediaStrings CheckBox Show media size distributions at recharge equilibrium on the MediaStrings page.
Options
ShowQFeed CheckBox QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
ShowQProd CheckBox QProd and associated tab pages (eg Sp) will become visible, showing the properties of the products.
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.

Mill page

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

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
VariableRates
HelpLink ButtonModelHelp.png Opens a link to this page using the system default web browser. Note: Internet access is required.
Mill
Diameter Input The inside liner diameter of the mill.
BellyLength Input The inside liner belly length of the mill, excluding cones.
TrunnionDiameter Input The inside liner trunnion diameter of the mill.
ConeAngle Input Angle of the feed and discharge end cones, measured as positive displacement from the vertical direction.
FracCS Input The fraction critical speed of the mill.
Grate
OpenAreaFrac Input Open area fraction of the grate.
PebblePortFrac / fp Input Pebble port area fraction.
PebblePortAperture / xp Input Pebble port aperture size.
GrateAperture / xg Input Grate aperture size.
FineSize / xm Input Fine size, size at which particles behave like water.
MeanRadialPosition / gamma Input Mean radial position of the grate apertures.
SlurryDischCoeff / k Input Slurry discharge coefficient.
Ball
BallLoad Input Ball load fraction.
BallSG Input Density (Specific Gravity) of ball media.
NumBallMeshSizes Input Number of ball mesh sizes below the top size, including the submesh.
BallTopSize Input Top size of new ball media.
Size Input / Display Ball sizing intervals.
Load Input Mass fraction retained of ball media in each ball sizing interval.
RFunction
RSize Display Spline knot positions.
RConst Input Values of at each spline knot position.
Other
ReferenceF80 Input Reference F80 size.
InternalMeshTopSize Input Top size of internal mesh series.
Voidage Input Volumetric fraction of void space in charge.
NetPowerAdjust Input Net Power Adjust factor of mill power equation.

Ore page

This page is used to define the comminution properties of SysCAD species with the size distribution quality in the project.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Ore
OreSpecific CheckBox
  • Ore-specific parameters, allows the ore breakage 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.
A Input / Display Impact ore breakage parameter.
b Input / Display Impact ore breakage parameter.
ta Input / Display Abrasion ore breakage parameter.
AGSAGVariableRates5.png AGSAGVariableRates6.png

Results page

This page is used to display the model results.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Results
Solver
Iterations Display Number of iterations to converge internal load solver.
IterationError Display Numerical approximation error of internal load solver.
Mill Properties
MillVolume Display Internal volume of the mill.
MillSpeed Display Rotational speed of the mill.
MillFeedRate / Feed.SLQv Display Volumetric feed rate of pulp into the mill.
Mill Contents
OreMass Display Mass of ore (solids with PSD) in the mill.
LiquidMass Display Mass of liquids in the mill.
BallMass Display Mass of ball media in the mill.
TotalChargeMass Display Total mass of ore, liquids and balls in the mill.
VolTotalLoad Display Volumetric fraction of mill volume of total charge (ore, liquids, balls and void space).
Mill Discharge
m1 Display Parameter of the Austin mill holdup relationship.
m2 Display Parameter of the Austin mill holdup relationship.
dMax Display Maximum discharge rate of load volume through the grate.
Charge Properties
S20 Display Size of the top (largest) 20% of the load.
ChargeDensity Display Density of the charge.
U Display Fraction of charge void space filled with slurry.
ThetaShoulder Display Angular position of the charge shoulder.
ThetaTue Display Angular position of the charge toe.
ChargeSurfaceRadius Display Radius of the inner charge surface.
Power
NoLoadPower Display No-load power draw of the mill.
NetPower Display Net power draw of the mill.
GrossPower Display Gross power draw of the mill.

RiDi page

This page displays the breakage and discharge rates for each size interval computed by the model.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Rates
Size Display Size of each interval in internal mesh series.
MeanSize Display Geometric mean size of each interval in internal mesh series.
R Display Value of breakage rate, , for each size interval, for each ore species.
D Display Value of discharge rate, , for each size interval.
Ecs Display Value of the specific comminution energy for each size interval.

Load page

This page displays information about the balls, solids and liquids that currently comprise the mill load.

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.
Load
SolidMass / SMt Display The mass of solids with the SysCAD size distribution property currently in the mill.
LiquidMass / LMt Display The mass of liquids plus solids without the SysCAD size distribution property currently in the mill.
BallMass / BMt Display The mass of ball media in the mill.
Size Display Size of each interval in the external mesh series.
MeanSize Display Geometric mean size of each interval in the external mesh series.
Load Display The mass of solids with the SysCAD size distribution property currently in the mill, by size and species.

Power page

This optional page displays the inputs and results for alternative mill power models. The page is only visible if PowerModels is selected on the MD_Mill page.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Power
HoggFuerstenau CheckBox Shows inputs and results for tumbling mill power calculations using the Hogg and Fuerstenau method.
MorrellC CheckBox Shows inputs and results for tumbling mill power calculations using the Morrell Continuum method.
MorrellE CheckBox Shows inputs and results for tumbling mill power calculations using the Morrell Empirical method.
MorrellD CheckBox Shows inputs and results for tumbling mill power calculations using the Morrell Discrete Shell method.
HildenPowell CheckBox Shows inputs and results for tumbling mill power calculations using the Hilden and Powell method.

MediaStrings page

This page displays the inputs and results for grinding mill media string calculations. The page is only visible if MediaStrings is selected on the MD_Mill page.

MediaTraj page

This page displays the inputs and results for tumbling mill media trajectory calculations. The page is only visible if MediaTrajectory is selected on the MD_Mill page.

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.

See also

References

  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.
  2. Leung, K., Morrison, R.D. and Whiten, W.J., 1987. An Energy Based Ore Specific Model for Autogenous and Semi-autogenous Grinding, Copper 87, Vina del Mar, Vol. 2, pp 71 - 86
  3. Morrell, S. and Morrison, R.D., 1996. AG and SAG mill circuit selection and design by simulation. In International Conference on Autogenous and Semiautogenous Grinding Technology (Vol. 2, pp. 769-790).
  4. Morrell, S., Valery, W., Banini, G. and Latchireddi, S., 2001. Developments in AG/SAG mill modelling. Proceedings of Autogenous and Semiautogenous Grinding Technology, Vancouver, pp.71-84.