View source for Rod Mill (Lynch)

== Description ==

This article describes an implementation of the '''Lynch''' (1977) rod mill model.{{Lynch (1977)}}{{Napier-Munn et al. (1996)}}

== Model theory ==

[[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.

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 <math>\sqrt{2}</math> 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 <math>\sqrt{2}</math> 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. [[Ball Mill (Perfect Mixing)|Perfect Mixing model]]). 

Thus, the product from a rod mill after <math>v</math> stages of breakage is determined from:{{Napier-Munn et al. (1996)}}

:<math>
\begin{align}
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{align}
</math>
where:
* <math>i</math> is the index of the size interval, <math>i = \{1,2,\dots,n\}</math>, <math>n</math> is the number of size intervals
* <math>j</math> is the index of the stage of breakage, <math>j = \{1,2,\dots,v\}</math>, <math>v</math> is the number of stages of breakage, <math>v \leq n</math>
* <math>f_i</math> is the mass flow rate of solids of size interval <math>i</math> in the mill feed
* <math>p_{i,j}</math> is the mass flow rate of solids of size interval <math>i</math> in the product of breakage stage <math>j</math>
* <math>p_{i,1} = f_i</math>, i.e. the mill feed
* <math>p_{i,v}</math> is the mass flow rate of solids of size interval <math>i</math> in the product from the final stage of breakage, i.e. the mill discharge
* <math>q_{i,j}</math> is the solids contents of size interval <math>i</math> in breakage stage <math>j</math>
* <math>S_{i,j}</math> is the selection function, the fraction of particles of size <math>i</math> which are broken in stage <math>j</math>
* <math>A_{i,k}</math> is the appearance function, the distribution of particle mass arising from the breakage of a parent particle in size interval <math>i</math> into progeny of size interval <math>k</math>
* <math>C_{i,j}</math> is the classification function, the fraction of particles of size <math>i</math> in breakage stage <math>j</math> 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 <math>S_{i,j} = S_{i-j+1,1}</math> and <math>C_{i,j} = C_{i-j+1,1}</math> for <math>j \leq i \leq n</math>.

=== 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:

:<math>
\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}
</math>

=== Selection ===

[[File:RodMillLynch8.png|thumb|400px|Figure 2. Rod mill selection function (after Napier-Munn et al, 1996).{{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:

:<math>
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}
</math>

where:

:<math>
\begin{array}{l}
K_1 = {\rm XC}, & K_2 = & \dfrac{1-{\rm IN}}{{\rm SL}}\\
\end{array}
</math>

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>{\rm SL}</math> is the slope of the selection function linear equation (/mm)
* <math>{\rm IN}</math> is the intercept of the selection function linear equation (frac)
* <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)

=== Appearance ===

{{Model theory (Text, Ball Mill, Perfect Mixing, Appearance)}}

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

:<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:
* <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>Q_{\rm M,F}</math> is the mass flow rate of particles in the feed (t/h)
* the <math>\rm Orig</math> and <math>\rm Sim</math> 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:

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

where:

:<math>
\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}
</math>

and:
* <math>D</math> is mill diameter (m)
* <math>L</math> is mill length (m)
* <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)

==== Ore hardness ====

The selection function is scaled for ore hardness by:

:<math>S_{\rm Sim} = S_{\rm Orig} \cdot f_{\rm WI}</math>

where:

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

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

=== Internal mesh series ===

{{Model theory (Text, Ball Mill, Perfect Mixing, Internal mesh)|Lynch rod}}

=== Multi-component modelling ===

{{Model theory (Text, Ball Mill, Perfect Mixing, Multi-component)|Lynch rod|selection function}}

== Excel ==

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

<syntaxhighlight lang="vb">=mdUnit_RodMill_Lynch(Parameters as Range, Size as Range, MillFeed as Range, SelectionParams as Range, Classification as Range, Appearance as Range, WorkIndexSim As Range)</syntaxhighlight>

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>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}
</math>

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

</math>

where:
* <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)

=== Results ===

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

:<math>
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}
</math>

where:
* <math>\text{Mill speed}</math> is the [[Tumbling Mill (Speed)|rotational rate of the mill]] (rpm)
* <math>Q_{\rm M,P}</math> 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.

{|
|- style="vertical-align:top;"
| [[File:RodMillLynch1.png|left|frame|Figure 1. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]] 
[[File:RodMillLynch4.png|left|frame|Figure 4. Example showing the outline of the '''Results''' (light blue frame) array in Excel.]] 
|| [[File:RodMillLynch2.png|left|frame|Figure 2. Example showing the selection of the '''Size''' (red frame) and '''MillFeed''' (purple frame).]] || [[File:RodMillLynch3.png|left|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.

{{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;" |''Lynch''

{{SysCAD (Text, Help Link)}}

|-
! colspan="3" style="text-align:left;" |''Requirements''
{{SysCAD (Text, NumParallelUnits)}}
|-
! colspan="3" style="text-align:left;" |''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.
|-
! colspan="3" style="text-align:left;" |''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.
|-
! colspan="3" style="text-align:left;" |''Results''
|-
|MillSpeed
|style="background: #eaecf0" | Display
|Rotational speed of simulated mill.
|-
|NumBreakageStages
|style="background: #eaecf0" | Display
|Number of breakage stages in the mill.
|}

{{SysCAD (Page, Ball Mill, Perfect Mixing, Ore)|method=Rod}}

==== Selection page ====

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

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''Scaling''
|-
|Diameter
|style="background: #eaecf0" | Display
|Value of the mill diameter-length factor for rate scaling.
|-
|LoadFraction
|style="background: #eaecf0" | Display
|Value of the load fraction factor for rate scaling.
|-
|FracCS
|style="background: #eaecf0" | Display
|Value of the fraction critical speed factor for rate scaling.
|-
|F90
|style="background: #eaecf0" | Display
|Value of the F90 factor for rate scaling.
|-
|WorkIndex
|style="background: #eaecf0" | Display
|Value of the Work Index factor of each ore species for rate scaling.
|-
! colspan="3" style="text-align:left;" |''Selection''
|-
|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.
|-
| Orig
|style="background: #eaecf0" {{!}} Display
| Value of original selection function at at size interval.
|-
| Selection
|style="background: #eaecf0" {{!}} Display
| Value of selection function of the simulated mill for each size interval, for each ore species.
|}

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