AG/SAG Mill (Variable Rates)

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).

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

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 ($i$) x column ($j$) format:

${\mathit {Parameters}}={\begin{bmatrix}D{\text{ (m)}}\\L{\text{ (m)}}\\D_{\rm {t}}{\text{ (m)}}\\\alpha _{c}{\text{ (deg.)}}\\\phi {\text{ (frac)}}\\A_{\rm {OF}}{\text{ (m}}^{\text{2}}{\text{)}}\\f_{\rm {p}}{\text{ (m}}^{2}{\text{/m}}^{2}{\text{)}}\\x_{\rm {p}}{\text{ (mm)}}\\x_{\rm {g}}{\text{ (mm)}}\\x_{\rm {m}}{\text{ (mm)}}\\\gamma {\text{ (m/m)}}\\k{\text{ (-)}}\\J_{\rm {B}}{\text{ (v/v)}}\\\rho _{\rm {B}}{\text{ (t/m}}^{3}{\text{)}}\\d_{\rm {B,Top}}{\text{ (mm)}}\\F_{80}{\text{ (mm)}}\\d_{1}{\text{ (mm)}}\\\varepsilon {\text{ (v/v)}}\\(Q_{\rm {M,F}})_{\rm {L}}{\text{ (t/h)}}\\\rho _{\rm {L}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\u\\v\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {Size}}={\begin{bmatrix}{\hat {d}}_{1}{\text{ (mm)}}\\\vdots \\{\hat {d}}_{r}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {MillNewFeed}}={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{r1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{rm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {OreSG}}={\begin{bmatrix}(\rho _{\rm {S}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}},\;\;\;\;\;\;$

${\mathit {BallSizing}}={\begin{bmatrix}(d_{\rm {B}})_{1}{\text{ (mm)}}&({\mathit {MF}}_{\rm {B}})_{1}{\text{ (}}\%{\text{ w/w)}}\\\vdots &\vdots \\(d_{\rm {B}})_{s}{\text{ (mm)}}&({\mathit {MF}}_{\rm {B}})_{s}{\text{ (}}\%{\text{ w/w)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {RConst}}={\begin{bmatrix}{\mathit {RConst}}_{1}{\text{ (}}\ln {\rm {h^{-1}{\text{)}}}}\\\vdots \\{\mathit {RConst}}_{5}{\text{ (}}\ln {\rm {h^{-1}{\text{)}}}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {OreBreakageParams}}={\begin{bmatrix}A_{1}{\text{ (}}\%{\text{ w/w)}}&\dots &A_{m}{\text{ (}}\%{\text{ w/w)}}\\b_{1}&\dots &b_{m}\\(t_{\rm {a}})_{1}{\text{ (}}\%{\text{ w/w)}}&\dots &(t_{\rm {a}})_{m}{\text{ (}}\%{\text{ w/w)}}\\\end{bmatrix}},\quad {\mathit {MillRecycleFeed}}={\begin{bmatrix}(Q_{\rm {M,R}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,R}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,R}})_{r1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,R}})_{rm}{\text{ (t/h)}}\\\end{bmatrix}}^{*}$

where:

• $(Q_{\rm {M,F}})_{\rm {L}}$ is the mass flow feed rate of liquids into the mill (t/h)
• $\rho _{\rm {L}}$ is the Specific Gravity or density of liquids in the feed (- or t/m³)
• $u$ is an index of the Appearance function to view in the results
• $v$ is an index of the Appearance function to view in the results
• $m$ is the number of ore types
• $r$ is the number of intervals of the external mesh series
• $s$ is the number of intervals of the ball mesh series below the top size, including the submesh
• ${\hat {d}}_{i}$ is the size of the external square mesh interval that feed mass is retained on (mm)
• ${\hat {d}}_{i+1}<{\hat {d}}_{i}<{\hat {d}}_{i-1}$, i.e. descending size order from top size (${\hat {d}}_{1}$) to sub mesh (${\hat {d}}_{p}=0$)
• $(d_{\rm {B}})_{i}$ is the size of the square mesh interval that balls are retained on (mm)
• $({\mathit {MF}}_{\rm {B}})_{i}$ is the mass fraction of balls retained on ball mesh series interval $i$ (% w/w)
• $^{*}$ indicates the ${\mathit {MillRecycleFeed}}$ 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:

${\mathit {mdUnit\_AGSAG\_VariableRates}}={\begin{bmatrix}{\begin{bmatrix}{\text{Iterations}}\\{\text{Iteration error}}\\V{\text{ (m}}^{\text{3}}{\text{)}}\\{\text{Mill speed (rpm)}}\\Q_{\rm {V,F}}\\M_{\rm {S}}{\text{ (t)}}\\M_{\rm {L}}{\text{ (t)}}\\M_{\rm {B}}{\text{ (t)}}\\M{\text{ (t)}}\\J_{\rm {t}}{\text{ (v/v)}}\\U{\text{ (v/v)}}\\k_{\rm {g}}{\text{ (-)}}\\m_{1}{\text{ (-)}}\\m_{2}{\text{ (-)}}\\d_{\rm {max}}{\text{ (h}}^{-1}{\text{)}}\\S_{20}{\text{ (mm)}}\\E_{q}{\text{ (kWh)}}\\\rho _{\rm {c}}{\text{ (t/m}}^{3}{\text{)}}\\\theta _{\rm {S}}{\text{ (rad)}}\\\theta _{\rm {T}}{\text{ (rad)}}\\r_{\rm {i}}{\text{ (m)}}\\P_{\rm {NoLoad}}{\text{ (kW)}}\\P_{\rm {Net}}{\text{ (kW)}}\\P_{\rm {Gross}}{\text{ (kW)}}\\\end{bmatrix}}&{\begin{array}{cccccccc}{\begin{bmatrix}{\hat {d}}_{1}{\text{ (mm)}}\\\vdots \\{\hat {d}}_{q}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,P}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,P}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,P}})_{q1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,P}})_{qm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(M_{\rm {S}})_{11}{\text{ (t)}}&\dots &(M_{\rm {S}})_{1m}{\text{ (t)}}\\\vdots &\ddots &\vdots \\(M_{\rm {S}})_{q1}{\text{ (t)}}&\dots &(M_{\rm {S}})_{qm}{\text{ (t)}}\\\end{bmatrix}}&{\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\\\end{bmatrix}}&{\begin{bmatrix}D_{1}\left({\text{h}}^{\text{-1}}\right)\\\vdots \\D_{n}\left({\text{h}}^{\text{-1}}\right)\\\end{bmatrix}}&{\begin{bmatrix}R_{1}\left({\text{h}}^{\text{-1}}\right)\\\vdots \\R_{n}\left({\text{h}}^{\text{-1}}\right)\\\end{bmatrix}}&{\begin{bmatrix}(E_{\rm {cs}})_{1}{\text{ (kWh/t)}}\\\vdots \\(E_{\rm {cs}})_{n}{\text{ (kWh/t)}}\\\end{bmatrix}}&{\begin{bmatrix}A_{u1}{\text{ (frac)}}\\\vdots \\A_{un}{\text{ (frac)}}\\\end{bmatrix}}&{\begin{bmatrix}A_{v1}{\text{ (frac)}}\\\vdots \\A_{vn}{\text{ (frac)}}\\\end{bmatrix}}\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\\end{array}}\end{bmatrix}}$

where:

• ${\text{Iterations}}$ is the number of internal computation steps required to converge the load
• ${\text{Iteration error}}$ is the numerical error of the converged load approximation
• $Q_{\rm {V,F}}$ is the flow rate of pulp into the mill (m³/h)
• ${\text{Mill speed}}$ is the rotational rate of the mill (rpm)
• $M_{\rm {S}}$ is the mass of ore solids in the mill (t)
• $M_{\rm {L}}$ is the mass of liquids in the mill (t)
• $M_{\rm {B}}$ is the mass of balls in the mill (t)
• $M$ is the total mass of ore, liquids and balls in the mill (t)
• $Q_{\rm {M,P}}$ is product mass flow rate (t/h)
• ${\bar {d}}_{i}$ 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		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 ${\mathit {RConst}}$ 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.

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, $R_{i}$, for each size interval, for each ore species.
D	Display	Value of discharge rate, $D_{i}$, 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		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.

See also

• Tumbling Mill (Slurry Flow)
• Tumbling Mill (Power, Morrell Continuum)
• Crusher (Whiten)