View source for 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).{{Napier-Munn et al. (1996)}}{{Leung et al. (1987)}}{{Morrell and Morrison (1996)}}

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

== Model theory ==

{{Under construction|section}}

== Excel ==

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

<syntaxhighlight lang="vb">=mdUnit_AGSAG_VariableRates(Parameters as Range, Size as Range, MillNewFeed as Range, MillRecycleFeed as Range, OreSG as Range, BallSizing as Range, RSize as Range, RConst as Range, OreBreakageParams as Range)</syntaxhighlight>

{{Excel (Text, Help, No Arguments)}}

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>Parameters=
\begin{bmatrix}
D\text{ (m)}\\
L\text{ (m)}\\
D_{\rm t}\text{ (m)}\\
\alpha_{c}\text{ (deg.)}\\
\phi\text{ (frac)}\\
A\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}\text{ (mm)}\\
(F_{80})_{\rm Ref}\text{ (mm)}\\
(d_1)_{\rm Int}\text{ (mm)}\\
\varepsilon\text{ (v/v)}\\
(Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\
\rho_{\rm L}\text{ (t/m}^{\text{3}}\text{)}\\
p\\
q\\
\end{bmatrix},\;\;\;\;\;\;

Size = \begin{bmatrix}
d_{1}\text{ (mm)}\\ 
\vdots\\ 
d_n\text{ (mm)}\\ 
\end{bmatrix},\;\;\;\;\;\;

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})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,F})_{nm}\text{ (t/h)}\\ 
\end{bmatrix},\;\;\;\;\;\;

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})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,R})_{nm}\text{ (t/h)}\\ 
\end{bmatrix}

</math>

:<math>
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},\;\;\;\;\;\;

BallSizing = \begin{bmatrix}
(\mathit{MF}_{\rm B})_{d_{\rm B}/\sqrt{2}}\text{ (}%\text{ w/w)}\\
(\mathit{MF}_{\rm B})_{d_{\rm B}/(2\sqrt{2}})\text{ (}%\text{ w/w)}\\
(\mathit{MF}_{\rm B})_{d_{\rm B}/(3\sqrt{2}})\text{ (}%\text{ w/w)}\\
(\mathit{MF}_{\rm B})_{d_{\rm B}/(4\sqrt{2}})\text{ (}%\text{ w/w)}\\
\end{bmatrix},\;\;\;\;\;\;

RSize= \begin{bmatrix}
d_{1}\text{ (mm)}\\ 
\vdots\\ 
d_{k}\text{ (mm)}\\ 
\end{bmatrix},\;\;\;\;\;\;

RConst= \begin{bmatrix}
RConst_1\\ 
\vdots\\ 
RConst_k\\ 
\end{bmatrix}
</math>

:<math>OreBreakageParams=
\begin{bmatrix}
A_1 & \dots & A_k\\
b_1 & \dots & b_k\\
(t_{\rm a})_1 & \dots & (t_{\rm a})_k\\
\end{bmatrix}   
</math>

where:
* <math>D</math> is the diameter of the mill inside the liners (m)
* <math>L</math> is the length of the belly section of the mill (m)
* <math>D_{\rm t}</math> is diameter of the discharge trunnion (m)
* <math>\alpha_{\rm c}</math> is angle between the cone end surface and the vertical direction (deg)
* <math>\phi</math> is the fraction critical speed of the mill (frac)
* <math>A</math> is the open area fraction of the discharge grate (frac)
* <math>f_{\rm p}</math> is the pebble port area fraction (frac)
* <math>x_{\rm p}</math> is the size of the pebble port aperture (mm)
* <math>x_{\rm g}</math> is the size of the grate aperture (mm)
* <math>x_{\rm m}</math> is the fine size (mm)
* <math>\gamma</math> is the mean radial position of the grate apertures (m/m)
* <math>k</math> is the slurry discharge coefficient
* <math>J_{\rm B}</math> is the ball load volume fraction (v/v)
* <math>\rho_{\rm B}</math> is the density or specific gravity of the ball media (t/m<sup>3</sup> or -)
* <math>d_{\rm B}</math> is ball top size (mm)
* <math>(F_{80})_{\rm Ref}</math> is the reference ''F<sub>80</sub>'' size (mm)
* <math>(d_1)_{\rm Int}</math> is the internal mesh top size (mm)
* <math>\varepsilon</math> is the charge void fraction (v/v)
* <math>(Q_{\rm M,F})_{\rm L}</math> is the mass flow feed rate of liquids into the mill (t/h)
* <math>\rho_{\rm L}</math> is the Specific Gravity or density of liquids in the feed (- or t/m<sup>3</sup>)
* <math>p</math> is an index of the Appearance function to view in the results
* <math>q</math> is an index of the Appearance function to view in the results
* <math>n</math> is the number of intervals
* <math>m</math> is the number of ore types
* <math>d_i</math> is the size of the square mesh interval that feed mass is retained on (mm)
* <math>d_{i+1}<d_i<d_{i-1}</math>, i.e. descending size order from top size (<math>d_{1}</math>) to sub mesh (<math>d_{n}=0</math> mm)
* <math>Q_{\rm M,F}</math> is new feed mass flow rate (t/h)
* <math>Q_{\rm M,R}</math> is recycle feed mass flow rate (t/h)
* <math>\rho_{\rm S}</math> is the density or specific gravity of solids (t/m<sup>3</sup> or -)
* <math>\mathit{MF}_{\rm B}</math> is the mass fraction retained of balls (% w/w)
* <math>k</math> is the number of breakage rate knots
* <math>RConst</math> is a breakage rate constant
* <math>A</math> (%), <math>b</math>, and <math>t_{\rm a}</math> are ore breakage parameters

=== Results ===

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

:<math>
\mathit{mdUnit\_AGSAG\_VariableRates} = 
\begin{bmatrix}

\begin{bmatrix}
\text{Iterations}\\
\text{Iteration error}\\
\text{Mill volume (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)}\\
\mathit{EL}\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}
d_1\text{ (mm)}\\
\vdots\\
d_n\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})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,P})_{nm}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{bmatrix}
(M_{\rm S})_{11}\text{ (t/h)} & \dots & (M_{\rm S})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(M_{\rm S})_{n1}\text{ (t/h)} & \dots & (M_{\rm S})_{nm}\text{ (t/h)}\\
\end{bmatrix}

& 

\begin{bmatrix}
d_1\text{ (mm)}\\
\vdots\\
d_{31}\text{ (mm)}
\end{bmatrix}

&
\begin{bmatrix}
\bar{d}_1\text{ (mm)}\\
\vdots\\
\bar{d}_{31}\text{ (mm)}\\
\end{bmatrix}

&

\begin{bmatrix}
D_{1}\left(\text{h}^\text{-1}\right)\\
\vdots\\
D_{31}\left(\text{h}^\text{-1}\right)\\
\end{bmatrix}

&

\begin{bmatrix}
R_{1}\left(\text{h}^\text{-1}\right)\\
\vdots\\
R_{31}\left(\text{h}^\text{-1}\right)\\
\end{bmatrix}

&

\begin{bmatrix}
(E_{\rm cs})_{1}\text{ (kWh/t)}\\
\vdots\\
(E_{\rm cs})_{31}\text{ (kWh/t)}\\
\end{bmatrix}

&

\begin{bmatrix}
A_{p,1}\text{ (frac)}\\
\vdots\\
A_{p,31}\text{ (frac)}\\
\end{bmatrix}

&

\begin{bmatrix}
A_{q,1}\text{ (frac)}\\
\vdots\\
A_{q,31}\text{ (frac)}\\
\end{bmatrix}\\

& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\

\end{array}

\end{bmatrix}
</math>

where:
* <math>\text{Iterations}</math> is the number of internal computation steps required to converge the load
* <math>\text{Iteration error}</math> is the numerical error of the converged load approximation
* <math>Q_{\rm V,F}</math> is the flow rate of pulp into the mill (m<sup>3</sup>/h)
* <math>\text{Mill volume}</math> is the total volume inside the mill, calculated as the sum of a cylinder and two frustums (m<sup>3</sup>)
* <math>\text{Mill speed}</math> is the [[Tumbling Mill (Speed)|rotational rate of the mill]] (rpm)
* <math>M_{\rm S}</math> is the mass of ore solids in the mill (t)
* <math>M_{\rm L}</math> is the mass of liquids in the mill (t)
* <math>M_{\rm B}</math> is the mass of balls in the mill (t)
* <math>M</math> is the total mass of ore, liquids and balls in the mill (t)
* <math>J_{\rm t}</math> is the charge volume fraction (v/v)
* <math>U</math> is the void fill fraction (v/v)
* <math>k_{\rm g}</math> is the coarse factor (-)
* <math>m_1</math> is a parameter of the Austin mill holdup equation
* <math>m_2</math> is a parameter of the Austin mill holdup equation
* <math>d_{\rm Max}</math> is maximum discharge rate from the mill (h<sup>-1</sup>)
* <math>S_{20}</math> is the geometric mean size of the top 20% of the load (mm)
* <math>\mathit{EL}</math> is the energy level in the mill (kWh)
* <math>\rho_{\rm c}</math> is the charge density (t/m<sup>3</sup>)
* <math>\theta_{\rm S}</math> is angular position of the charge shoulder (rad)
* <math>\theta_{\rm T}</math> is angular position of the charge toe (rad)
* <math>r_{\rm i}</math> is the charge surface radius (m)
* <math>P_{\rm NoLoad}</math> is the no-load power of the mill (kW)
* <math>P_{\rm Net}</math> is the net power of the mill (kW)
* <math>P_{\rm Gross}</math> is the gross power of the mill (kW)
* <math>Q_{\rm M,P}</math> is product mass flow rate (t/h)
* <math>\bar{d}_i</math> is the [[Conversions|geometric mean size]] of the internal mesh series interval that mass is retained on (mm)
* <math>D</math> is the discharge rate (h<sup>-1</sup>)
* <math>R</math> is the breakage rate (h<sup>-1</sup>)
* <math>E_{\rm cs}</math> is specific comminution energy (kWh/t)
* <math>A</math> is the Appearance function (frac)

=== Example ===

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

{| 
|- style="vertical-align:top;"
| [[File:AGSAGVariableRates1.png|left|frame|Figure 1. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]] 
| [[File:AGSAGVariableRates2.png|left|frame|Figure 2. Example showing the selection of the '''Size''' (red frame), '''OreSG''' (pink frame), '''MillNewFeed''' (purple frame) and '''MillRecycleFeed''' (green frame) arrays in Excel.]] 
|}
{| 
|- style="vertical-align:top;"
| [[File:AGSAGVariableRates3.png|left|frame|Figure 3. Example showing the selection of the '''BallSizing''' (brown frame), '''RSize''' (teal frame), '''RConst''' (blue frame), and '''OreBreakageParams''' (red frame) arrays in Excel.]] 
| [[File:AGSAGVariableRates4.png|left|frame|Figure 8. 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.

{{SysCAD (Page, Mill, DLL*Mill)|PowerModels=true|MediaTraj=true|MediaStrings=true}}

==== Mill page ====

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

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''VariableRates''

{{SysCAD (Text, Help Link)}}

|-
! colspan="3" style="text-align:left;" |''Requirements''
{{SysCAD (Text, NumParallelUnits)}}
|-
! colspan="3" style="text-align:left;" |''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.
|-
! colspan="3" style="text-align:left;" |''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.
|-
! colspan="3" style="text-align:left;" |''Ball''
|-
|BallLoad
|Input
|Ball load fraction.
|-
|BallSG
|Input
|Density (Specific Gravity) of ball media.
|-
|BallTopSize
|Input
|Top size of new ball media.
|-
|Size
|style="background: #eaecf0" | Display
|Ball sizing intervals.
|-
|Load
|style="background: #eaecf0" | Display
|Mass fraction retained of ball media in each ball sizing interval.
|-
! colspan="3" style="text-align:left;" |''RFunction''
|-
|Size
|Input
|Spline knot size positions.
|-
| RConst
|Input
|Values of <math>RConst</math> at each spline knot position.
|-
! colspan="3" style="text-align:left;" |''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.

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''Ore''
|- style="vertical-align:top;"
|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.
|-
|colspan="3" | [[File:AGSAGVariableRates5.png]] [[File:AGSAGVariableRates6.png]]
|}

==== Results page ====

This page is used to display the model results.

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''Results''
|-
! colspan="3" style="text-align:left;" |''Solver''
|- 
|Iterations
|style="background: #eaecf0" | Display
|Number of iterations to converge internal load solver.
|- 
|IterationError
|style="background: #eaecf0" | Display
|Numerical approximation error of internal load solver.
|-
! colspan="3" style="text-align:left;" |''Mill Properties''
|- 
|MillVolume
|style="background: #eaecf0" | Display
|Internal volume of the mill.
|- 
|MillSpeed
|style="background: #eaecf0" | Display
|Rotational speed of the mill.
|- 
|MillFeedRate / Feed.SLQv
|style="background: #eaecf0" | Display
|Volumetric feed rate of pulp into the mill.
|-
! colspan="3" style="text-align:left;" |''Mill Contents''
|- 
|OreMass
|style="background: #eaecf0" | Display
|Mass of ore (solids with PSD) in the mill.
|- 
|LiquidMass
|style="background: #eaecf0" | Display
|Mass of liquids in the mill.
|- 
|BallMass
|style="background: #eaecf0" | Display
|Mass of ball media in the mill.
|- 
|TotalChargeMass
|style="background: #eaecf0" | Display
|Total mass of ore, liquids and balls in the mill.
|- 
|VolTotalLoad
|style="background: #eaecf0" | Display
|Volumetric fraction of mill volume of total charge (ore, liquids, balls and void space).
|-
! colspan="3" style="text-align:left;" |''Mill Discharge''
|- 
|m1
|style="background: #eaecf0" | Display
|Parameter of the Austin mill holdup relationship.
|- 
|m2
|style="background: #eaecf0" | Display
|Parameter of the Austin mill holdup relationship.
|- 
|dMax
|style="background: #eaecf0" | Display
|Maximum discharge rate of load volume through the grate.
|-
! colspan="3" style="text-align:left;" |''Charge Properties''
|- 
|S20
|style="background: #eaecf0" | Display
|Size of the top (largest) 20% of the load.
|- 
|ChargeDensity
|style="background: #eaecf0" | Display
|Density of the charge.
|- 
|U
|style="background: #eaecf0" | Display
|Fraction of charge void space filled with slurry.
|- 
|ThetaShoulder
|style="background: #eaecf0" | Display
|Angular position of the charge shoulder.
|- 
|ThetaTue
|style="background: #eaecf0" | Display
|Angular position of the charge toe.
|- 
|ChargeSurfaceRadius
|style="background: #eaecf0" | Display
|Radius of the inner charge surface.
|-
! colspan="3" style="text-align:left;" |''Power''
|- 
|NoLoadPower
|style="background: #eaecf0" | Display
|No-load power draw of the mill.
|- 
|NetPower
|style="background: #eaecf0" | Display
|Net power draw of the mill.
|- 
|GrossPower
|style="background: #eaecf0" | 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.

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''Rates''
|-
|Size
|style="background: #eaecf0" | Display
|Size of each interval in internal mesh series.
|-
|MeanSize
|style="background: #eaecf0" | Display
|Geometric mean size of each interval in internal mesh series.
|-
| R
|style="background: #eaecf0" | Display
| Value of breakage rate, <math>R_i</math>, for each size interval, for each ore species. 
|-
| D
|style="background: #eaecf0"; | Display
| Value of discharge rate, <math>D_i</math>, for each size interval. 
|-
| Ecs
|style="background: #eaecf0"; | 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.

{{SysCAD_Table_Header}}

{{SysCAD (Text, Distribution)}}

|-
! colspan="3" style="text-align:left;" |''Load''
|-
|SolidMass / SMt
|style="background: #eaecf0" | Display
|The mass of solids with the SysCAD size distribution property currently in the mill.
|-
|LiquidMass / LMt
|style="background: #eaecf0" | Display
|The mass of liquids plus solids without the SysCAD size distribution property currently in the mill.
|-
|BallMass / BMt
|style="background: #eaecf0" | Display
|The mass of ball media in the mill.
|-
|Size
|style="background: #eaecf0" | Display
|Size of each interval in the external mesh series.
|-
|MeanSize
|style="background: #eaecf0" | Display
|Geometric mean size of each interval in the external mesh series.
|-
|Load
|style="background: #eaecf0" | Display
|The mass of solids with the SysCAD size distribution property currently in the mill, by size and species.
|}

{{SysCAD (Page, Tumbling Mill, Power)|modelpage={{Name (Text, Company Name)|nospace=1}}*Mill|HF=true|MorrellC=true|MorrellE=true|MorrellD=true|HildenPowell=true}}

{{SysCAD (Page, Tumbling Mill, MediaStrings)|modelpage={{Name (Text, Company Name)|nospace=1}}*Mill}}

{{SysCAD (Page, Tumbling Mill, MediaTraj)|modelpage={{Name (Text, Company Name)|nospace=1}}*Mill}}

{{SysCAD (Page, About)}}

== References ==

[[Category:Excel]]
[[Category:SysCAD]]