Stirred Mill (Perfect Mixing, Dynamic)

Description

This article describes a dynamic implementation of the Perfect Mixing vertical stirred mill model outlined by Napier-Munn et al. (1996).

The dynamic version uses the same underlying theory and structure as the steady-state Perfect Mixing stirred mill model. For a full description of the steady-state model, see Stirred Mill (Perfect Mixing).

Model theory

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Excel

The Perfect Mixing stirred mill model is not implemented in Excel in dynamic form for practical purposes. Excel is not an ideal platform for dynamic simulation and SysCAD (or similar) is preferred.

The dynamic model is, however, included in Excel in a run-to-steady-state mode where all feed and input parameters are fixed and time steps are progressed until the computed load and discharge stabilises.

This mode is useful for extracting separated $R_{i}$ and $D_{i}$ functions from steady-state data such as plant surveys or other model calibrations (including the steady-state Perfect Mixing stirred mill model).

The run-to-steady-state dynamic Perfect Mixing stirred mill model may be invoked from the Excel formula bar with the following function call:

=mdUnit_StirredMill_PerfectMixingRiDi(Parameters as Range, Size as Range, MillFeed as Range, OreSG As Range, Appearance as Range, WorkIndexSim as Range, RKnotPositions as Range, RKnotsOrig as Range, Classification as Range, Optional ScalingExponents as Range)

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:

$Parameters={\begin{bmatrix}D{\text{ (m)}}\\H{\text{ (m)}}\\(H_{\rm {b}})_{\rm {Orig}}{\text{ (m)}}\\(D_{\rm {b}})_{\rm {Orig}}{\text{ (mm)}}\\(D_{\rm {s}})_{\rm {Orig}}{\text{ (m)}}\\(N_{\rm {s}})_{\rm {Orig}}{\text{ (rpm)}}\\S_{\rm {Orig}}\\T_{\rm {Orig}}\\(H_{\rm {b}})_{\rm {Sim}}{\text{ (m)}}\\(D_{\rm {b}})_{\rm {Sim}}{\text{ (mm)}}\\(D_{\rm {s}})_{\rm {Sim}}{\text{ (m)}}\\(N_{\rm {s}})_{\rm {Sim}}{\text{ (rpm)}}\\S_{\rm {Sim}}\\T_{\rm {Sim}}\\{\rm {WI}}_{\rm {Orig}}{\text{ (kWh/t)}}\\(Q_{\rm {M,F}})_{\rm {L}}{\text{ (t/h)}}\\\rho _{\rm {L}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\{\text{Classification method}}\\x_{\rm {m}}{\text{ (mm)}}\\{\text{Discharge type}}\\{\text{User overflow volume (m}}^{\text{3}}{\text{)}}\\\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;MillFeed={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;OreSG={\begin{bmatrix}(\rho _{\rm {S}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}}$

$Appearance={\begin{bmatrix}{\begin{bmatrix}A_{1}{\text{ (frac)}}\\\vdots \\A_{31}{\text{ (frac)}}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}A_{1}{\text{ (frac)}}\\\vdots \\A_{31}{\text{ (frac)}}\\\end{bmatrix}}_{m}\end{bmatrix}},\;\;\;\;\;\;{\rm {WI}}_{\rm {Sim}}={\begin{bmatrix}{\rm {WI}}_{1}{\text{ (kWh/t)}}&\dots &{\rm {WI}}_{m}{\text{ (kWh/t)}}\\\end{bmatrix}},\;\;\;\;\;\;R/D^{*}KnotPositions={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{k}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;R/D^{*}KnotOrig={\begin{bmatrix}\ln \left({\frac {R}{D^{*}}}\right)_{1}\\\vdots \\\ln \left({\frac {R}{D^{*}}}\right)_{k}\\\end{bmatrix}}$

$Classification={\begin{cases}{\begin{bmatrix}C_{1}{\text{ (frac)}}\\\vdots \\C_{31}{\text{ (frac)}}\\\end{bmatrix}}&{\mbox{if Classification method}}=0{\mbox{ (User)}}\\x_{\rm {g}}&{\mbox{if Classification method}}=1{\mbox{ (Leung)}}\\\end{cases}}$

${\mathit {ScalingExponents}}={\begin{bmatrix}e_{1}\\e_{2}\\e_{3}\\e_{4}\\e_{5}\\e_{6}\\e_{7}\\\end{bmatrix}}^{*}$

where:

• $(Q_{\rm {M,F}})_{\rm {L}}$ is the mass flow feed rate of liquids into the mill (t/h)
• $\rho _{\rm {B}}$ is the Specific Gravity or density of the media in the mill (- or t/m³)
• $\rho _{\rm {L}}$ is the Specific Gravity or density of liquids in the feed (- or t/m³)
• $m$ is the number of ore types
• $k$ is the number of breakage rate per discharge rate knots
• $d_{i}$ is the size of the square mesh interval that feed mass is retained on (mm)
• $d_{i+1}<d_{i}<d_{i-1}$, i.e. descending size order from top size ($d_{1}$) to sub mesh ($d_{n}=0$ mm)
• $Q_{\rm {M,F}}$ is the mass flow rate of particles in the feed (t/h)
• $\rho _{\rm {S}}$ is the Specific Gravity or density of solids (- or t/m³)
• ${\text{Classification method }}$ is the method used to defined the classification-by-size to discharge, 0 = User-defined partition or 1 = Leung method
• ${\text{Discharge type}}$ is discharge configuration, 0 = Overflow discharge at mill height, 1 = Overflow discharge at user-defined slurry filling volume
• ${\text{User overflow volume}}$ is the user-specified slurry filling volume at which overflow commences (if ${\text{Discharge type}}=1$) (m³)
• $^{*}$ indicates the ${\mathit {ScalingExponents}}$ array is optional, the default values are used if omitted.

Results

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

$mdUnit\_StirredMill\_PerfectMixing\_RiDi={\begin{bmatrix}{\begin{bmatrix}Q_{\rm {V}}{\text{ (m}}^{\text{3}}{\text{/h)}}\\V_{\rm {Mill}}{\text{ (m}}^{\text{3}}{\text{)}}\\V_{\rm {Media}}\\V_{\rm {Slurry}}\\{\rm {LF}}{\text{ (v/v)}}\\\tau {\text{ (min)}}\\{\text{R/D* factor (frac)}}\\f_{\rm {Ds}}{\text{ (-)}}\\f_{\rm {Hb}}{\text{ (-)}}\\f_{\rm {Ns}}{\text{ (-)}}\\f_{\rm {S}}{\text{ (-)}}\\f_{\rm {T}}{\text{ (-)}}\\f_{\rm {Db}}{\text{ (-)}}\\f_{\rm {WI}}{\text{ (-)}}\\{\text{Iterations}}\\dt{\text{ (s)}}\\d_{\rm {max}}{\text{ (h}}^{\text{-1}}{\text{)}}\\{\text{Ore mass (t)}}\\{\text{Liquid mass (t)}}\\{\text{Ball mass (t)}}\\{\text{Slurry in mill (m}}^{\text{3}}{\text{)}}\\{\text{Max. slurry in charge (m}}^{\text{3}}{\text{)}}\\{\text{Max. slurry in mill (OF) (m}}^{\text{3}}{\text{)}}\\(Q_{\rm {M,P}})_{\rm {L}}{\text{ (t/h)}}\\\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_{11}{\text{ (t/h)}}&\dots &M_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\M_{n1}{\text{ (t/h)}}&\dots &M_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{31}{\text{ (mm)}}\\\end{bmatrix}}&{\begin{bmatrix}R_{11}\left({\text{h}}^{\text{-1}}\right)&\dots &R_{1m}\left({\text{h}}^{\text{-1}}\right)\\\vdots &\ddots &\vdots \\R_{31,1}\left({\text{h}}^{\text{-1}}\right)&\dots &R_{31m}\left({\text{h}}^{\text{-1}}\right)\\\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}C_{1}{\text{ (frac)}}\\\vdots \\C_{31}{\text{ (frac)}}\\\end{bmatrix}}&{\begin{bmatrix}\ln R_{11}&\dots &\ln R_{1m}\\\vdots &\ddots &\vdots \\\ln R_{k1}&\dots &\ln R_{km}\\\end{bmatrix}}\\&&&&&&&\\&&&&&&&{\begin{bmatrix}R_{11}\left({\text{h}}^{\text{-1}}\right)&\dots &R_{1m}\left({\text{h}}^{\text{-1}}\right)\\\vdots &\ddots &\vdots \\R_{k1}\left({\text{h}}^{\text{-1}}\right)&\dots &R_{km}\left({\text{h}}^{\text{-1}}\right)\\\end{bmatrix}}\\&&&&&&&\\&&&&&&&{\begin{bmatrix}(Factor_{\rm {WI}})_{1}&\dots &(Factor_{\rm {WI}})_{m}\end{bmatrix}}\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\&&&&&&\\\end{array}}\end{bmatrix}}$

where:

• ${\rm {LF}}$ is the media load fraction (v/v), defined as the fraction of mill volume occupied by media, including void spaces, i.e. ${\rm {LF}}=H_{\rm {b}}{\big /}H$
• ${\text{R/D* factor}}=D_{i}^{*}/D_{i}$ is the discharge rate scaling factor
• $Q_{\rm {M,P}}$ is the mass flow rate of particles in the mill product (t/h)
• ${\bar {d}}_{i}$ is the geometric mean size of the internal mesh series interval that mass is retained on (mm)
• ${\text{Iterations}}$ is the number of time steps required to reach steady-state
• $dt$ is the size of the discretised time step calculated by the model, $\Delta t$ (s)
• ${\text{Ore mass}}$ is the total mass of ore in the mill at steady-state (t)
• ${\text{Liquid mas}}$ is the mass of liquids in the mill at steady-state (t)
• ${\text{Ball mass}}$ is the mass of balls in the mill at steady-state (t)
• ${\text{Slurry in mill}}$ is the volume of slurry in the mill at steady-state (m³)
• ${\text{Max. slurry in charge}}$ is the maximum volume of slurry that can occupy the charge void space before (m³)
• ${\text{Max. slurry in mill (OF)}}$ is the maximum volume of slurry in the mill before overflow commences (m³)
• $(Q_{\rm {M,P}})_{\rm {L}}$ is the discharge mass flow rate of liquids from the mill (t/h)
• $M$ is the mass of solids in the mill (t)

Example

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

Figure 4. Example showing the selection of the Parameters (blue frame) array in Excel.

Figure 5. Example showing the selection of the Appearance (pink frame), RKnotPositions (teal frame), RKnotsOrig (blue frame), WorkIndexSim (dark red frame), Classification (Leung, $x_{\rm {g}}$) (red frame) arrays in Excel.

Figure 6. Example showing the selection of the Size (red frame), OreSG (green frame) and MillFeed (purple frame) arrays in Excel.

Figure 7. Example showing the outline of the Results (light blue frame) array in Excel.

SysCAD

The SysCAD interface for Dynamic mode is described below. For steady-state, see Stirred Mill (Perfect Mixing).

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.
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
PerfectMixing
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Mode	Steady State	The steady-state Perfect Mixing stirred mill model is used in SysCAD Dynamic simulation mode used instead of the dynamic Perfect Mixing model described here. This approach may be useful when very large SysCAD step sizes are used and steady-state operation can be assumed between each time step.
	Dynamic	The dynamic Perfect Mixing stirred mill model described here is used to determine the mill product size distribution. Different parameters can be used for different solids.
MinSubSteps	Input	The user-specified minimum number of internal models steps taken per SysCAD step.
SubSteps	Display	The actual number of internal models steps taken per SysCAD step. May be affected by breakage/discharge rates or the user-specified MinSubSteps parameter.
DischargeType	Overflow	The maximum slurry volume in the mill before overflowing is calculated by the model.
	Overflow (User)	The maximum slurry volume in the mill before overflowing is specified by the user.
Stirred
MediaStringsP50	CheckBox	Only visible if the MediaStrings option is checked. Replaces the MediaSize.Sim user defined value with the MediaSize (All) P50 value from the MediaStrings page. The value of MediaSize.Orig should be determined on the same basis for correct scaling of a changed media charge.
UserExponents	Checkbox	Indicates whether to use user-defined values for the scaling exponents.
MillDiameter	Input	The inside liner diameter of the simulated mill.
MillHeight	Input	The height of the simulated mill.
MediaHeight	Input	Height of the media charge of the original and simulated mills.
MediaDiameter	Input	Median diameter of the media in the original and simulated mills.
StirrerDiameter	Input	Diameter of the stirrers of the original and simulated mills.
StirrerSpeed	Input	Rotational speed of the stirrers of the original and simulated mills.
StirrerStarts	Input	Number of starts of the (screw) stirrers of the original and simulated mills.
StirrerTurns	Input	Number of turns per start of the (screw) stirrers of the original and simulated mills.
WorkIndex	Input	Bond Ball Work Index of ore in the original mill.
Exponent	Input	Only visible if UserExponents is selected Allows user to specify values of the scaling exponent for each original/simulated parameter pair.
RFunction
NumSplineKnots	Input	Number of spline knots for the $R_{i}$ function.
Size	Input	Spline knot size positions.
Ln(R)	Input	Values of $\ln R$ at each spline knot position.
Results
MillVolume	Display	Volume inside the mill.
MediaVolume	Display	Volume occupied by the media in the mill, excluding void space.
SlurryVolume	Display	Volume of slurry in the mill, including slurry in media void space and above media charge.
LoadFraction	Display	Volume of media charge, including void space and excluding stirrer volume, as a fraction of mill volume, excluding stirrer volume.
ResidenceTime	Display	Mean residence time of slurry in the mill.

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
Appearance
DefaultAppearance		Sets all species to the the default Kelsall Appearance function.
OreSpecific	CheckBox	Ore-specific parameters, allows the Appearance data 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.
Appearance	Input	User-specified Appearance function data for all species with size distribution property.
WorkIndex
WorkIndex.Sim	Input	Bond Ball Work Index data for all species with size distribution property.

RiDi page

This page displays the scaling factors and breakage rate per discharge rate for each size interval computed by the Perfect Mixing model.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Scaling
StirrerDiameter	Display	Value of the stirrer diameter factor for rate scaling.
MediaHeight	Display	Value of the media height factor for rate scaling.
StirrerSpeed	Display	Value of the stirrer speed factor for rate scaling.
StirrerStarts	Display	Value of the stirrer starts factor for rate scaling.
StirrerTurns	Display	Value of the stirrer turns per start factor for rate scaling.
MediaSize	Display	Value of the media size factor for rate scaling.
WorkIndex	Display	Value of the Work Index factor of each ore species for rate scaling.
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.
C	Display	Value of classification function, $C_{i}$, for each size interval.
D	Display	Value of discharge rate, $D_{i}$, 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
Filling
SLCapacity / SLVtCap	Input / Display	The maximum volume of slurry the mill can contain before overflow.
SLCharge / SLVtCharge	Display	Only appears if DischargeType is 'Overflow'. The maximum volume of slurry that can fill the charge void space.
SLVolume / SLVt	Display	The total volume of slurry currently in the mill.
SLLevel / SLLvl	Display	The current slurry volume (SLVolume) as a fraction of the maximum slurry volume before overflow (SLCapacity).
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 internal mesh series.
MeanSize	Display	Geometric mean size of each interval in internal mesh series.
Load	Display	The mass of solids with the SysCAD size distribution property currently in the mill, by size and species.

Content, Sp, Ec, Sz and MSz pages

These pages display the standard SysCAD Material Content, Species Content and Size pages for the current mill load

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
Nitta	CheckBox	Shows inputs and results for stirred mill power calculations using the Nitta method.
Heath	CheckBox	Shows inputs and results for stirred mill power calculations using the Heath 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.

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

• Steady-state Perfect Mixing stirred mill model
• Herbst-Fuerstenau mill model
• Stirred Mill (Power, Nitta)
• Stirred Mill (Power, Heath)