Rod Mill (Lynch): Difference between revisions

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== Model theory ==
== Model theory ==
{{Restricted content}}


[[File:RodMillLynch7.png|thumb|525px|Figure 1. Schematic diagram illustrating one stage of breakage (after Napier-Munn et al, 1996).{{Napier-Munn et al. (1996)}}]]
<hide>
[[File:RodMillLynch7.png|thumb|475px|Figure 1. Schematic diagram illustrating one stage of breakage (after Napier-Munn et al, 1996).{{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.{{Lynch (1977)}} The preferential selection manifests as a "screening" effect which restricts the movement of coarser particles towards the discharge.
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.{{Lynch (1977)}} The preferential selection manifests as a "screening" effect which restricts the movement of coarser particles towards the discharge.
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S_i =
S_i =
\begin{cases}
\begin{cases}
\mathit{SL} \cdot XC + \mathit{IN} & \bar d_i < K_1\\
{\rm SL} \cdot {\rm XC} +{\rm IN} & \bar d_i < K_1\\
\mathit{SL} \cdot \bar d_i + \mathit{IN} & K_1 \leq \bar d_i \leq K_2\\
{\rm SL} \cdot \bar d_i + {\rm IN} & K_1 \leq \bar d_i \leq K_2\\
1 & \bar d_i > K_2\\
1 & \bar d_i > K_2\\
\end{cases}
\end{cases}
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:<math>
:<math>
\begin{array}{l}
\begin{array}{l}
K_1 = \mathit{XC}, & K_2 = & \dfrac{1-\mathit{IN}}{\mathit{SL}}\\
K_1 = {\rm XC}, & K_2 = & \dfrac{1-{\rm IN}}{{\rm SL}}\\
\end{array}
\end{array}
</math>
</math>
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and:
and:
* <math>K_1</math> and <math>K_2</math> are particle sizes with similar meanings to the same parameters used by the [[Crusher (Whiten)|Whiten crusher]] model (mm)
* <math>K_1</math> and <math>K_2</math> are particle sizes with similar meanings to the same parameters used by the [[Crusher (Whiten)|Whiten crusher]] model (mm)
* <math>\mathit{SL}</math> is the slope of the selection function linear equation (/mm)
* <math>{\rm SL}</math> is the slope of the selection function linear equation (/mm)
* <math>\mathit{IN}</math> is the intercept of the selection function linear equation (frac)
* <math>{\rm IN}</math> is the intercept of the selection function linear equation (frac)
* <math>\mathit{XC}</math> is the point of intersection (mm)
* <math>{\rm XC}</math> is the point of intersection (mm)
* <math>\bar d_{i}</math> is the [[Conversions|geometric mean size]] of particles in size interval <math>i</math> (mm)
* <math>\bar d_{i}</math> is the [[Conversions|geometric mean size]] of particles in size interval <math>i</math> (mm)


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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:{{Napier-Munn et al. (1996)}}
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:{{Napier-Munn et al. (1996)}}


:<math>v_{\rm sim} = \left ( \dfrac{\mathit{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}}}</math>
:<math>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}}}</math>


where:
where:
* <math>\mathit{MC}</math> is the ''mill constant'', a constant that links the number of breakage stages to the mill feed rate
* <math>{\rm MC}</math> is the ''mill constant'', a constant that links the number of breakage stages to the mill feed rate
* <math>F_{90}</math> is 90% passing size in the mill feed (mm)
* <math>F_{90}</math> is 90% passing size in the mill feed (mm)
* <math>Q_{\rm M,F}</math> is the mass flow rate of particles in the feed (t/h)
* <math>Q_{\rm M,F}</math> is the mass flow rate of particles in the feed (t/h)
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The mill constant may be scaled by changes in the design or operating conditions of a mill to be simulated:
The mill constant may be scaled by changes in the design or operating conditions of a mill to be simulated:


:<math>\mathit{MC}_{\rm Sim} = \mathit{MC}_{\rm Orig} \cdot f_{\rm DL} \cdot f_{\rm LF} \cdot f_{\rm CS}</math>
:<math>{\rm MC}_{\rm Sim} = {\rm MC}_{\rm Orig} \cdot f_{\rm DL} \cdot f_{\rm LF} \cdot f_{\rm CS}</math>


where:
where:
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\begin{array}{l}
\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 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 - \mathit{LF}_{\rm Sim}}{1 - \mathit{LF}_{\rm Orig}} \right ) \left ( \dfrac{\mathit{LF}_{\rm Sim}}{\mathit{LF}_{\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 )\\
f_{\rm CS} = \left ( \dfrac{(\mathit{C}_{\rm s})_{\rm Sim}}{(\mathit{C}_{\rm s})_{\rm Orig}}  \right )\\
\end{array}
\end{array}
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* <math>D</math> is mill diameter (m)
* <math>D</math> is mill diameter (m)
* <math>L</math> is mill length (m)
* <math>L</math> is mill length (m)
* <math>\mathit{LF}</math> is load fraction, the load volume as a fraction of mill volume (v/v)
* <math>{\rm LF}</math> is load fraction, the load volume as a fraction of mill volume (v/v)
* <math>C_{\rm s}</math> is the fraction critical speed of the mill (frac)
* <math>C_{\rm s}</math> is the fraction critical speed of the mill (frac)


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where:
where:


:<math>f_{\rm WI} = \left ( \dfrac{\mathit{WI}_{\rm Sim}}{\mathit{WI}_{\rm Orig}} \right )^{0.8}</math>
:<math>f_{\rm WI} = \left ( \dfrac{{\rm WI}_{\rm Sim}}{{\rm WI}_{\rm Orig}} \right )^{0.8}</math>


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


=== Internal mesh series ===
=== Internal mesh series ===
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=== Multi-component modelling ===
=== Multi-component modelling ===


{{Model theory (Text, Ball Mill, Perfect Mixing, Multi-component)|Lynch rod|selection function}}
The original Lynch rod mill 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.
</hide>


== Excel ==
== Excel ==
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D_{\rm Orig}\text{ (m)}\\
D_{\rm Orig}\text{ (m)}\\
L_{\rm Orig}\text{ (m)}\\
L_{\rm Orig}\text{ (m)}\\
\mathit{LF}_{\rm Orig}\text{ (v/v)}\\
{\rm LF}_{\rm Orig}\text{ (v/v)}\\
(C_{\rm s})_{\rm Orig}\text{ (frac)}\\
(C_{\rm s})_{\rm Orig}\text{ (frac)}\\
\mathit{WI}_{\rm Orig}\text{ (kWh/t)}\\
{\rm WI}_{\rm Orig}\text{ (kWh/t)}\\
\mathit{MC}_{\rm Orig}\\
{\rm MC}_{\rm Orig}\\
(F_{90})_{\rm Orig}\text{ (mm)}\\
(F_{90})_{\rm Orig}\text{ (mm)}\\
D_{\rm Sim}\text{ (m)}\\
D_{\rm Sim}\text{ (m)}\\
L_{\rm Sim}\text{ (m)}\\
L_{\rm Sim}\text{ (m)}\\
\mathit{LF}_{\rm Sim}\text{ (v/v)}\\
{\rm LF}_{\rm Sim}\text{ (v/v)}\\
(C_{\rm s})_{\rm Sim}\text{ (frac)}\\
(C_{\rm s})_{\rm Sim}\text{ (frac)}\\
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;
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\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


SelectionParams= \begin{bmatrix}
SelectionParams=  
\mathit{XC}\text{ (mm)}\\
\begin{bmatrix}
\mathit{IN}\text{ (mm)}\\
{\rm XC}\text{ (mm)}\\
\mathit{SL}\\
{\rm IN}\text{ (mm)}\\
{\rm SL}\\
\end{bmatrix}
\end{bmatrix}
</math>
</math>
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\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


\mathit{WI}_{\rm Sim}= \begin{bmatrix}
{\rm WI}_{\rm Sim}= \begin{bmatrix}
\mathit{WI}_{1}\text{ (kWh/t)} & \dots & \mathit{WI}_m\text{ (kWh/t)}\\  
{\rm WI}_{1}\text{ (kWh/t)} & \dots & {\rm WI}_m\text{ (kWh/t)}\\  
\end{bmatrix}\;\;\;\;\;\;
\end{bmatrix}\;\;\;\;\;\;


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\begin{bmatrix}
\begin{bmatrix}
\text{Mill speed (rpm)}\\
\text{Mill speed (rpm)}\\
\mathit{MC}_{\rm Sim}\\
{\rm MC}_{\rm Sim}\\
v_{\rm Sim}\\
v_{\rm Sim}\\
f_{\rm DL}\text{ (-)}\\
f_{\rm DL}\text{ (-)}\\
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{{SysCAD (Text, Help Link)}}
{{SysCAD (Text, Help Link)}}


|-
! colspan="3" style="text-align:left;" |''Requirements''
{{SysCAD (Text, NumParallelUnits)}}
|-
|-
! colspan="3" style="text-align:left;" |''Scaling''
! colspan="3" style="text-align:left;" |''Scaling''
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|}
|}


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


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


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


{{SysCAD (Page, About)}}
{{SysCAD (Page, About)}}

Latest revision as of 02:12, 9 December 2024

Description

This article describes an implementation of the Lynch (1977) rod mill model.[1][2]

Model theory

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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 () x column () format:

where:

  • is the number of ore types
  • 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)

Results

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

where:

  • is the rotational rate of the mill (rpm)
  • 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.
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
Lynch
HelpLink ButtonModelHelp.png Opens a link to this page using the system default web browser. Note: Internet access is required.
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

ButtonSet.png

Sets the Classification function to the default values.
DefaultAppearance

ButtonSetAll.png

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.

RodMillLynch5.png RodMillLynch6.png

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 ButtonLicensingHelp.png Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information ButtonCopyToClipboard.png 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 ButtonBrowse.png 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.

References

  1. ↑ Lynch, A.J., 1977. Mineral Crushing and Grinding Circuits: Their Simulation, Optimisation, Design and Control. Elsevier, Amsterdam.
  2. ↑ Napier-Munn, 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.
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