Rod Mill (Lynch)

Description

This article describes an implementation of the Lynch (1977) rod mill model.

Model theory

Figure 1. Schematic diagram illustrating one stage of breakage (after Napier-Munn et al, 1996).

Lynch (1977) describes a rod mill as a plug flow device which preferentially selects coarser particles for breakage as size reduction progresses down the mill. The preferential selection manifests as a "screening" effect which restricts the movement of coarser particles towards the discharge.

To quantify this observation, Lynch defines a "stage of breakage" as the length of a mill required to completely eliminate all particles from the largest ${\sqrt {2}}$ size interval in the feed. The product from such a stage becomes the feed for a subsequent stage, eliminating particles from the next size interval in the ${\sqrt {2}}$ series, and so on.

The size reduction in a rod mill may then be described as a series of selection, breakage and classification processes occurring in stages down the length of the mill, as illustrated schematically in Figure 1.

The rod mill size reduction model was originally formulated as a sequence of matrix operations. However, this implementation adopts an equivalent algebraic population balance formulation similar to other models (e.g. Perfect Mixing model).

Thus, the product from a rod mill after $v$ stages of breakage is determined from:

${\begin{aligned}p_{i,j}=&(1-C_{i,j})\cdot q_{i,j}\\q_{i,j}=&{\dfrac {(1-S_{i,j})\cdot p_{i,j-1}+\sum \limits _{k=1}^{i}{A_{i,k}S_{k,j}(C_{k,j}q_{k,j}+p_{k,j-1})}}{1-C_{i,j}+S_{i,j}C_{i,j}}}\\\\\end{aligned}}$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $j$ is the index of the stage of breakage, $j=\{1,2,\dots ,v\}$, $v$ is the number of stages of breakage, $v\leq n$
• $f_{i}$ is the mass flow rate of solids of size interval $i$ in the mill feed
• $p_{i,j}$ is the mass flow rate of solids of size interval $i$ in the product of breakage stage $j$
• $p_{i,1}=f_{i}$, i.e. the mill feed
• $p_{i,v}$ is the mass flow rate of solids of size interval $i$ in the product from the final stage of breakage, i.e. the mill discharge
• $q_{i,j}$ is the solids contents of size interval $i$ in breakage stage $j$
• $S_{i,j}$ is the selection function, the fraction of particles of size $i$ which are broken in stage $j$
• $A_{i,k}$ is the appearance function, the distribution of particle mass arising from the breakage of a parent particle in size interval $i$ into progeny of size interval $k$
• $C_{i,j}$ is the classification function, the fraction of particles of size $i$ in breakage stage $j$ which are prevented from entering the next stage of breakage.

As a model simplification, Lynch recommends the selection, appearance and classification functions are maintained constant relative to the top size of the current stage of breakage. Therefore $S_{i,j}=S_{i-j+1,1}$ and $C_{i,j}=C_{i-j+1,1}$ for $j\leq i\leq n$.

Classification

The classification function determines the fraction of particles which are prevented from entering the next stage of breakage, i.e. that are recycled back to the feed of the current stage of breakage.

The classification function adopts a simple geometric progression in order to satisfy Lynch's requirement that the top size of a breakage stage is completely eliminated. Thus, for the first stage of breakage:

${\begin{array}{l}C_{j=1}={\begin{bmatrix}1\\0.5\\0.25\\0.125\\0.0625\\\vdots \\{\dfrac {1}{2(n-1)}}\end{bmatrix}}&\implies &C_{i,1}={\begin{cases}1&i=1\\{\dfrac {1}{2(i-1)}}&2\leq i\leq n\\\end{cases}}\end{array}}$

Selection

Figure 2. Rod mill selection function (after Napier-Munn et al, 1996).

The selection function determines the fraction of particles of a given size which are subject to breakage events.

A simple linear equation is used to define the selection function, as illustrated in Figure 2. Mathematically, the selection function equation is described by:

$S_{i}={\begin{cases}{\rm {SL}}\cdot {\rm {XC}}+{\rm {IN}}&{\bar {d}}_{i}<K_{1}\\{\rm {SL}}\cdot {\bar {d}}_{i}+{\rm {IN}}&K_{1}\leq {\bar {d}}_{i}\leq K_{2}\\1&{\bar {d}}_{i}>K_{2}\\\end{cases}}$

where:

${\begin{array}{l}K_{1}={\rm {XC}},&K_{2}=&{\dfrac {1-{\rm {IN}}}{\rm {SL}}}\\\end{array}}$

and:

• $K_{1}$ and $K_{2}$ are particle sizes with similar meanings to the same parameters used by the Whiten crusher model (mm)
• ${\rm {SL}}$ is the slope of the selection function linear equation (/mm)
• ${\rm {IN}}$ is the intercept of the selection function linear equation (frac)
• ${\rm {XC}}$ is the point of intersection (mm)
• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)

Appearance

The appearance function describes the mass-by-size distribution of progeny particles resulting from the breakage of parent particles.

The appearance function may be specified for a particular ore. Alternatively, the default Broadbent-Callcott appearance function may be used, which is defined as:

$A_{ij}={\frac {1-\exp \left(-{\dfrac {d_{i}}{d_{j}}}\right)}{1-\exp(-1)}}$

where $d_{i}$ is the breakage product particle size and $d_{j}$ is the original particle size.

Scaling

Stages of breakage

Napier-Munn et al. (1996) suggests the number of stages of breakage of a mill to be simulated may be scaled from an original, fitted mill model by the relation:

$v_{\rm {sim}}=\left({\dfrac {{\rm {MC}}_{\rm {Sim}}}{Q_{\rm {M,F}}}}\right)^{\frac {2}{3}}+{\dfrac {\ln \left({\dfrac {(F_{90})_{\rm {Orig}}}{(F_{90})_{\rm {Sim}}}}\right)}{\ln {\sqrt {2}}}}$

where:

• ${\rm {MC}}$ is the mill constant, a constant that links the number of breakage stages to the mill feed rate
• $F_{90}$ is 90% passing size in the mill feed (mm)
• $Q_{\rm {M,F}}$ is the mass flow rate of particles in the feed (t/h)
• the ${\rm {Orig}}$ and ${\rm {Sim}}$ subscripts refer to the original and simulated mills, respectively.

The the number of stages of breakage may be a non-integer value, in which case linear interpolation between stages is used to compute the mill product.

Mill constant

The mill constant may be scaled by changes in the design or operating conditions of a mill to be simulated:

${\rm {MC}}_{\rm {Sim}}={\rm {MC}}_{\rm {Orig}}\cdot f_{\rm {DL}}\cdot f_{\rm {LF}}\cdot f_{\rm {CS}}$

where:

${\begin{array}{l}f_{\rm {DL}}=\left({\dfrac {D_{\rm {Sim}}}{D_{\rm {Orig}}}}\right)^{2.5}\cdot \left({\dfrac {L_{\rm {Sim}}}{L_{\rm {Orig}}}}\right),&f_{\rm {LF}}=\left({\dfrac {1-{\rm {LF}}_{\rm {Sim}}}{1-{\rm {LF}}_{\rm {Orig}}}}\right)\left({\dfrac {{\rm {LF}}_{\rm {Sim}}}{{\rm {LF}}_{\rm {Orig}}}}\right),&f_{\rm {CS}}=\left({\dfrac {({\mathit {C}}_{\rm {s}})_{\rm {Sim}}}{({\mathit {C}}_{\rm {s}})_{\rm {Orig}}}}\right)\\\end{array}}$

and:

• $D$ is mill diameter (m)
• $L$ is mill length (m)
• ${\rm {LF}}$ is load fraction, the load volume as a fraction of mill volume (v/v)
• $C_{\rm {s}}$ is the fraction critical speed of the mill (frac)

Ore hardness

The selection function is scaled for ore hardness by:

$S_{\rm {Sim}}=S_{\rm {Orig}}\cdot f_{\rm {WI}}$

where:

$f_{\rm {WI}}=\left({\dfrac {{\rm {WI}}_{\rm {Sim}}}{{\rm {WI}}_{\rm {Orig}}}}\right)^{0.8}$

and ${\rm {WI}}$ is the Bond Rod Work Index of the ore (kWh/t).

Internal mesh series

The Lynch rod mill model is formulated internally with a geometric progression of 31 mesh sizes at ${\sqrt {2}}$ intervals. Feed and product size fractions are automatically converted to and from the internal mesh series during model computation. The ${\sqrt {2}}$ size intervals allow the appearance function to be specified as a one-dimensional matrix, rather than the two dimensional form defined above, since

$A_{ij}=A_{i-j}$

when the intervals are so spaced.

Multi-component modelling

The original Lynch rod model formulation only considered the properties of a single ore type.

This implementation applies different appearance functions and selection function scaling factors to separate population balance computations for each ore type in the feed.

Excel

The Lynch rod mill model may be invoked from the Excel formula bar with the following function call:

=mdUnit_RodMill_Lynch(Parameters as Range, Size as Range, MillFeed as Range, SelectionParams as Range, Classification as Range, Appearance as Range, WorkIndexSim 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:

$Parameters={\begin{bmatrix}D_{\rm {Orig}}{\text{ (m)}}\\L_{\rm {Orig}}{\text{ (m)}}\\{\rm {LF}}_{\rm {Orig}}{\text{ (v/v)}}\\(C_{\rm {s}})_{\rm {Orig}}{\text{ (frac)}}\\{\rm {WI}}_{\rm {Orig}}{\text{ (kWh/t)}}\\{\rm {MC}}_{\rm {Orig}}\\(F_{90})_{\rm {Orig}}{\text{ (mm)}}\\D_{\rm {Sim}}{\text{ (m)}}\\L_{\rm {Sim}}{\text{ (m)}}\\{\rm {LF}}_{\rm {Sim}}{\text{ (v/v)}}\\(C_{\rm {s}})_{\rm {Sim}}{\text{ (frac)}}\\\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;MillFeed={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;SelectionParams={\begin{bmatrix}{\rm {XC}}{\text{ (mm)}}\\{\rm {IN}}{\text{ (mm)}}\\{\rm {SL}}\\\end{bmatrix}}$

$Classification={\begin{bmatrix}C_{1,1}{\text{ (frac)}}\\\vdots \\C_{31,1}{\text{ (frac)}}\\\end{bmatrix}},\;\;\;\;\;\;Appearance={\begin{bmatrix}{\begin{bmatrix}A_{1}{\text{ (frac)}}\\\vdots \\A_{31}{\text{ (frac)}}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}A_{1}{\text{ (frac)}}\\\vdots \\A_{31}{\text{ (frac)}}\\\end{bmatrix}}_{m}\end{bmatrix}},\;\;\;\;\;\;{\rm {WI}}_{\rm {Sim}}={\begin{bmatrix}{\rm {WI}}_{1}{\text{ (kWh/t)}}&\dots &{\rm {WI}}_{m}{\text{ (kWh/t)}}\\\end{bmatrix}}\;\;\;\;\;\;$

where:

• $m$ is the number of ore types
• $d_{i}$ 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)

Results

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

$mdUnit\_RodlMill\_Lynch={\begin{bmatrix}{\begin{bmatrix}{\text{Mill speed (rpm)}}\\{\rm {MC}}_{\rm {Sim}}\\v_{\rm {Sim}}\\f_{\rm {DL}}{\text{ (-)}}\\f_{\rm {LF}}{\text{ (-)}}\\f_{\rm {CS}}{\text{ (-)}}\\f_{F_{90}}{\text{ (-)}}\\\end{bmatrix}}&{\begin{array}{cccccc}{\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,P}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,P}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,P}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,P}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{31}{\text{ (mm)}}\\\end{bmatrix}}&{\begin{bmatrix}(S_{1,1})_{\rm {Orig}}{\text{ (frac)}}\\\vdots \\(S_{31,1})_{\rm {Orig}}{\text{ (frac)}}\\\end{bmatrix}}&{\begin{bmatrix}{\begin{bmatrix}S_{1,1}{\text{ (frac)}}\\\vdots \\S_{31,1}{\text{ (frac)}}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}S_{1,1}{\text{ (frac)}}\\\vdots \\S_{31,1}{\text{ (frac)}}\\\end{bmatrix}}_{m}\end{bmatrix}}&{\begin{array}{c}{\big [}(f_{\rm {WI}})_{1}&\dots &(f_{\rm {WI}})_{m}{\big ]}\\\\\\\end{array}}\\\\\\\\\\\end{array}}\end{bmatrix}}$

where:

• ${\text{Mill speed}}$ is the rotational rate of the mill (rpm)
• $Q_{\rm {M,P}}$ is the mass flow rate of particles in the mill product (t/h)

Example

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

Figure 1. Example showing the selection of the Parameters (blue frame) array in Excel. Figure 4. Example showing the outline of the Results (light blue frame) array in Excel.

Figure 2. Example showing the selection of the Size (red frame) and MillFeed (purple frame).

Figure 3. Example showing the selection of the SelectionParams (green frame), Classification (pink frame), Appearance (brown frame), and WorkIndexSim (teal frame) arrays 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.
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
Lynch
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: 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.
Scaling
Diameter	Input	The inside liner diameter of the original and simulated mills.
BellyLength	Input	The inside liner belly length of the simulated mill.
LoadFrac	Input	The volumetric load fraction of the original and simulated mills.
FracCS	Input	The fraction critical speed of the original and simulated mills.
WorkIndex	Input	Bond Rod Work Index of ore in the original mill.
MillConstant	Input	Mill constant of original and simulated mills.
F90	Input	Size passing 90% of feed in original and simulated mills.
Selection
SizeConstant / XC	Input	Size Constant value of the selection function.
Intercept / IN	Input	Intercept value of the selection function.
Slope / SL	Input	Slope value of the selection function.
Results
MillSpeed	Display	Rotational speed of simulated mill.
NumBreakageStages	Display	Number of breakage stages 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
DefaultClassification		Sets the Classification function to the default values.
DefaultAppearance		Sets all species to the the default Broadbent-Callcott 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 Rod Work Index data for all species with size distribution property.

Selection page

This page displays the selection function values computed for the Lynch rod mill model.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Scaling
Diameter	Display	Value of the mill diameter-length factor for rate scaling.
LoadFraction	Display	Value of the load fraction factor for rate scaling.
FracCS	Display	Value of the fraction critical speed factor for rate scaling.
F90	Display	Value of the F90 factor for rate scaling.
WorkIndex	Display	Value of the Work Index factor of each ore species for rate scaling.
Selection
Size	Display	Size of each interval in internal mesh series.
MeanSize	Display	Geometric mean size of each interval in internal mesh series.
Orig	Display	Value of original selection function at at size interval.
Selection	Display	Value of selection function of the simulated mill for each size interval, for each ore 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.

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.