Stirred Mill (Perfect Mixing, Dynamic)
Description
This article describes a dynamic implementation of the Perfect Mixing vertical stirred mill model outlined by NapierMunn et al. (1996).^{[1]}
The dynamic version uses the same underlying theory and structure as the steadystate Perfect Mixing stirred mill model. For a full description of the steadystate model, see Stirred Mill (Perfect Mixing).
Model theory
The dynamic Perfect Mixing model is based on a population balance of particles entering the mill, breaking into smaller sizes, and discharging as product. For a mill operating in unsteadystate, the diagram in Figure 1 below represents the balance for a given size fraction:
The dynamic population balance is described mathematically as:^{[2]}
where:
 is the index of the size interval, , is the number of size intervals
 is the mass feed rate of solids in size interval
 is the mass product rate of solids in size interval
 is the mass of solids in the mill load in size interval
 is the breakage rate of solids in the mill load in size interval
 is the rate of discharge from the mill of solids in size interval
 is the Appearance function, the distribution of particle mass arising from the breakage of a parent particle in size interval into progeny of size interval
Unlike the steadystate version, the load component cannot be eliminated from the equation, nor can the and components be combined into a single term. Therefore, the breakage and discharge rates and must be specified separately as inputs to the dynamic model.
Finally, an unsteadystate mill simulation must also consider the retention of liquids in the load:
where:
 is the load mass of water in the mill
 is the mass feed rate of water into the mill
 is the discharge rate of water from the mill, normally assumed to equal the value of at the finest size interval.
Time step discretisation
The unsteadystate Perfect Mixing differential equation is numerically solved by a discretised time stepping approach (i.e. Euler's method). The change in load mass in a size fraction during a sufficiently small time increment is:
Similarly, for liquids:
The time stepping approach is a convenient numerical approximation to the solution of the unsteadystate Perfect Mixing population balance differential equation. The approach is, however, subject to several limitations:
 The mass of particles separately discharged from or broken out of a size interval in a time step cannot exceed the mass of particles actually present in that size interval.
 Similarly, the overall maximum discharge flow rate of pulp from the mill cannot exceed the total volume of pulp in the mill in a time step.
The time step size used internally by the model is automatically reduced to ensure the breakage, discharge and pulp flow rate limits per step are not exceeded. This is achieved by computing a number of sequential substeps at the reduced internal step size for each requested external step.
This is useful if the either a fixed time step specified by an application is too large (e.g. SysCAD) or a numerical solution is desired in as few steps as possible (e.g. Excel). The automatic time step adjustment is largely invisible to the user and manifests only as a slightly slower execution speed.
The calculated time step size may be overridden with a larger userspecified substep count if increased accuracy in the numerical approximation is desired.
Breakage rate
The breakage function, , is an input parameter for the dynamic Perfect Mixing model, replacing the term appearing in the steadystate version.
The breakage rate, , is affected by mill operating conditions as follows:
 is the stirrer diameter (m)
 is the stirrer speed (rpm)
 is the number of starts of the screw stirrer
 is the number of turns per start along the full height of the screw stirrer ()
 is the mean media diameter (mm)
Adopting the same scaling approach and nomenclature as the Perfect Mixing ball mill, the rate is scaled by the following relation:
The scaling factors are defined as:
where:
 the subscript refers to the original mill from which was derived
 the subscript refers to the mill being simulated (scaled)
 are scaling exponents, , , and
The Work Index scaling factor, , is retained from the Perfect Mixing ball mill model, where is the Bond Ball Work Index value of the ore (kWh/t).
The model user may optionally change the values of to suit specific test work results or operating data.
The discharge rate scaling factor, , is excluded from the breakage rate scaling term and is dealt with separately within the model (see Slurry filling and discharge, below).
Discharge rate
The classical Leung definition of discharge rate of solids from a perfectly mixed mill is:^{[3]}
where:
and:
 is the fraction of load presented to the mill discharge
 is the classification function, the fraction of particles of size reporting to the mill product
 is the geometric mean size of particles in size interval (mm)
 is the particle size below which all mass in the size interval reports to mill product, i.e. like water
 is the largest particle size which can report to mill product
Prior work suggests that internal classification within stirred mills may be represented by a Whiten efficiency curve (Hasan et al., 2016).^{[4]} Leung's equation is likely to be an adequate approximation of the same classification behaviour.
Appearance function
The Appearance function describes the massbysize distribution of progeny particles resulting from the breakage of parent particles.
The Appearance function may be specified for a particular ore.
Morrell et al. (1993) found the appearance functions generated for the Perfect Mixing ball mill model were too fine for stirred mills. They recommended either generating new appearance functions at onetenth of the standard breakage test input energy level, or using the Kelsall breakage function.^{[5]}
Internal mesh series
The Perfect Mixing mill model is formulated internally with a geometric progression of 31 mesh sizes at intervals. Feed and product size fractions are automatically converted to and from the internal mesh series during model computation. The size intervals allow the appearance function to be specified as a onedimensional matrix, rather than the two dimensional form defined above, since
when the intervals are so spaced.
Multicomponent modelling
The original Perfect Mixing model formulation only considered the properties of a single ore type.
This implementation applies different appearance functions and breakage rate scaling factors to separate population balance computations for each ore type in the feed.
Slurry filling and discharge
The unsteadystate population balance and liquid holdup models described above compute the quantity of slurry in the mill at each discrete time step.
Discharge flow does not commence from an initially empty stirred mill until the available internal volume is filled. The available internal volume consists of the grinding media interstices and any remaining space above the media charge.
The discharge rate, , whilst filling the mill is zero. During this period, the dynamic population balance reduces to a batch mill formulation (with or without feed), as outlined by Whiten (1974):^{[6]}
Discharge commences once the internal mill volume is filled with slurry. The volumetric discharge rate of pulp from the mill is then, for practical purposes, equal to the instantaneous volumetric feed rate.
The model computes the value of to ensure the total flow rate of water plus solids classified for discharge matches the required product pulp outflow rate (i.e. the feed rate) for the mill.
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 runtosteadystate 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 and functions from steadystate data such as plant surveys or other model calibrations (including the steadystate Perfect Mixing stirred mill model).
The runtosteadystate 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 () x column () format:
where:
 is the mass flow feed rate of liquids into the mill (t/h)
 is the Specific Gravity or density of the media in the mill ( or t/m^{3})
 is the Specific Gravity or density of liquids in the feed ( or t/m^{3})
 is the number of ore types
 is the number of breakage rate per discharge rate knots
 is the size of the square mesh interval that feed mass is retained on (mm)
 , i.e. descending size order from top size () to sub mesh ( mm)
 is the mass flow rate of particles in the feed (t/h)
 is the Specific Gravity or density of solids ( or t/m^{3})
 is the method used to defined the classificationbysize to discharge, 0 = Userdefined partition or 1 = Leung method
 is discharge configuration, 0 = Overflow discharge at mill height, 1 = Overflow discharge at userdefined slurry filling volume
 is the userspecified slurry filling volume at which overflow commences (if ) (m^{3})
 indicates the 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:
where:
 is the media load fraction (v/v), defined as the fraction of mill volume occupied by media, including void spaces, i.e.
 is the discharge rate scaling factor
 is the mass flow rate of particles in the mill product (t/h)
 is the geometric mean size of the internal mesh series interval that mass is retained on (mm)
 is the number of time steps required to reach steadystate
 is the size of the discretised time step calculated by the model, (s)
 is the total mass of ore in the mill at steadystate (t)
 is the mass of liquids in the mill at steadystate (t)
 is the mass of balls in the mill at steadystate (t)
 is the volume of slurry in the mill at steadystate (m^{3})
 is the maximum volume of slurry that can occupy the charge void space before (m^{3})
 is the maximum volume of slurry in the mill before overflow commences (m^{3})
 is the discharge mass flow rate of liquids from the mill (t/h)
 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.
SysCAD
The SysCAD interface for Dynamic mode is described below. For steadystate, 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. 
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 (steadystate 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 (steadystate or dynamic) is used to determine the mill product size distribution. Different parameters can be used for different solids.  
Mill (HerbstFuerstenau)  The HerbstFuerstenau 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 userspecified size (in the field SizeForPassingFracCalc) in each stream. 
Passes  Display  The userspecified (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.  
Requirements  
NumParallelUnits  Input  The number of parallel, identical units to simulate:

Mode  Steady State 

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 userspecified 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 userspecified 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.  
MediaStringsP50  CheckBox 

Stirred  
UserExponents  Checkbox  Indicates whether to use userdefined 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 

RFunction  
NumSplineKnots  Input  Number of spline knots for the function. 
Size  Input  Spline knot size positions. 
Ln(R)  Input  Values of 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 

Appearance  Input  Userspecified 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, , for each size interval, for each ore species. 
C  Display  Value of classification function, , for each size interval. 
D  Display  Value of discharge rate, , 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 AddOn.
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 AddOn. 
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, MDSysCAD Full or MDSysCAD 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
 Steadystate Perfect Mixing stirred mill model
 HerbstFuerstenau mill model
 Stirred Mill (Power, Nitta)
 Stirred Mill (Power, Heath)
References
 ↑ ^{1.0} ^{1.1} NapierMunn, 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.
 ↑ Valery Jnr, W. and Morrell, S., 1995. The development of a dynamic model for autogenous and semiautogenous grinding. Minerals engineering, 8(11), pp.12851297.
 ↑ Leung, K., Morrison, R.D. and Whiten, W.J., 1987. An Energy Based Ore Specific Model for Autogenous and Semiautogenous Grinding, Copper 87, Vina del Mar, Vol. 2, pp 71  86
 ↑ Hasan, M., Palaniandy, S., Hilden, M. and Powell, M., 2016. Investigating internal classification within gravity induced stirred mills. Minerals Engineering, 95, pp.513.
 ↑ ^{5.0} ^{5.1} Morrell, S., Sterns, U.J. and Weller, K.R., 1993. The application of population balance models to very fine grinding in tower mills. 18th International Mineral Processing Congress; 2328 May, 1993; Sydney, NSW. 1993, pp. 6166.
 ↑ Whiten, W.J., 1974. A matrix theory of comminution machines. Chemical Engineering Science, 29(2), pp.589599.