Mill (Herbst-Fuerstenau): Difference between revisions

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== Model theory ==
== Model theory ==


{{Restricted content}}
<hide>
The Herbst-Fuerstenau 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 steady-state, the diagram in Figure 1 below represents the balance for a given size fraction:
The Herbst-Fuerstenau 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 steady-state, the diagram in Figure 1 below represents the balance for a given size fraction:


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where:
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>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>f_i</math> is the mass fractions of particles in size interval <math>i</math> appearing in the mill feed (w/w)
* <math>f_i</math> is the mass fraction of particles in size interval <math>i</math> appearing in the mill feed (w/w)
* <math>p_i</math> is the mass fractions of particles in size interval <math>i</math> appearing in the mill product (w/w)
* <math>p_i</math> is the mass fraction of particles in size interval <math>i</math> appearing in the mill product (w/w)
* <math>S_i</math> is the selection function at size interval <math>i</math>
* <math>S_i</math> is the selection function at size interval <math>i</math>
* <math>b_{ij}</math> is the breakage function, the mass fraction of a broken particle of original size <math>j</math> appearing in size <math>i</math>  (w/w)
* <math>b_{ij}</math> is the breakage function, the mass fraction of a broken particle of original size <math>j</math> appearing in size <math>i</math>  (w/w)
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The product of the selection function and mean residence time, <math>S_i \tau</math>, was found to be related to the energy input to grinding by:
The product of the selection function and mean residence time, <math>S_i \tau</math>, was found to be related to the energy input to grinding by:


:<math>S_i \tau = S^E_i \left ( \frac{P}{W} \right ) = S^E_i \bar E</math>
:<math>S_i \tau = S^{\rm E}_i \left ( \frac{P}{W} \right ) = S^{\rm E}_i \bar E</math>


where:
where:
* <math>S^E_i</math> is the ''energy-specific'' selection function at size interval <math>i</math>
* <math>S^{\rm E}_i</math> is the ''energy-specific'' selection function at size interval <math>i</math>
* <math>P</math> is the power drawn by the mill (excluding no-load power) (kW)
* <math>P</math> is the power drawn by the mill (excluding no-load power) (kW)
* <math>W</math> is the mass flow rate of solids through the mill at steady-state (t/h)
* <math>W</math> is the mass flow rate of solids through the mill at steady-state (t/h)
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1 & i=j\\
1 & i=j\\
\\
\\
\sum\limits_{k=j}^{i-1}{\dfrac{b_{ik}S^E_k}{S^E_i - S^E_j}}T_{kj} & i>j\\
\sum\limits_{k=j}^{i-1}{\dfrac{b_{ik}S^{\rm E}_k}{S^{\rm E}_i - S^{\rm E}_j}}T_{kj} & i>j\\
\end{cases}
\end{cases}
</math>
</math>
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{\rm e}^{-S_i \bar E} & \text{for batch or plug flow grinding when }i=j\\
{\rm e}^{-S_i \bar E} & \text{for batch or plug flow grinding when }i=j\\
\\
\\
\dfrac{1}{\left (1 + \dfrac{S^E_i \bar E}{N} \right )^N} & \text{for continuous grinding when }i=j\\
\dfrac{1}{\left (1 + \dfrac{S^{\rm E}_i \bar E}{N} \right )^N} & \text{for continuous grinding when }i=j\\
\\
\\
0 & i \neq j\\
0 & i \neq j\\
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where <math>N</math> is the number of mixers-in-series approximating the residence time distribution.
where <math>N</math> is the number of mixers-in-series approximating the residence time distribution.


Different relations are available for specifying the form of the energy-specific selection (<math>S^E_i</math>) and breakage (<math>b_{ij}</math>) functions, which are described below.
Different relations are available for specifying the form of the energy-specific selection (<math>S^{\rm E}_i</math>) and breakage (<math>b_{ij}</math>) functions, which are described below.


=== Energy-specific selection function ===
=== Energy-specific selection function ===
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The energy-specific selection function associated with the Herbst and Fuerstenau model is:{{King (2012)}}
The energy-specific selection function associated with the Herbst and Fuerstenau model is:{{King (2012)}}


:<math>S^E_i = S^E_1 \exp \left ( \zeta_1 \ln{ \left ( \dfrac{\bar d_{i}}{d_{p1}} \right )} + \zeta_2 \left [ \ln{ \left ( \dfrac{\bar d_{i}}{d_{p1}} \right )} \right ] ^2  \right )</math>
:<math>S^{\rm E}_i = S^{\rm E}_1 \exp \left ( \zeta_1 \ln{ \left ( \dfrac{\bar d_{i}}{d_{{\rm p}1}} \right )} + \zeta_2 \left [ \ln{ \left ( \dfrac{\bar d_{i}}{d_{{\rm p}1}} \right )} \right ] ^2  \right )</math>


where:
where:
* <math>S^E_1</math> is the value of the energy-specific selection function (<math>S^E_i</math>) at a nominal reference particle size <math>\bar d_{p1}</math> (mm)
* <math>S^{\rm E}_1</math> is the value of the energy-specific selection function (<math>S^{\rm E}_i</math>) at a nominal reference particle size <math>\bar d_{{\rm p}1}</math> (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)
* <math>\zeta_1</math> and <math>\zeta_2</math> are coefficients that characterise the shape of the energy-specific selection function curve
* <math>\zeta_1</math> and <math>\zeta_2</math> are coefficients that characterise the shape of the energy-specific selection function curve


King (2012) notes the value of <math>d_{p1}</math> is often taken as the top size of the mesh series in use. However, a nominal size such as 1 mm may be a better choice, as the selection function parameters are then transferrable across different mesh series (where the top size may change).
King (2012) notes the value of <math>d_{{\rm p}1}</math> is often taken as the top size of the mesh series in use, although a nominal size of 1mm may also be an adequate choice.


==== Austin ====
==== Austin ====
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An expanded form of the Austin selection function is available:{{Sepulveda (2005)}}
An expanded form of the Austin selection function is available:{{Sepulveda (2005)}}


:<math>S^E_i = \left( \dfrac{1}{1 + \dfrac{\alpha_2}{\alpha_1}} \right ) \left ( \dfrac{\alpha_{0} \bar {d_i}^{\alpha_{1}}}{1 + \left ( \dfrac{\bar d_i}{d_{crit}} \right )^{\alpha_2}} + \alpha_{02} \bar {d_i}^{\alpha_{12}} \right )</math>
:<math>S^{\rm E}_i = \left( \dfrac{1}{1 + \dfrac{\alpha_2}{\alpha_1}} \right ) \left ( \dfrac{\alpha_{0} \bar {d_i}^{\alpha_{1}}}{1 + \left ( \dfrac{\bar d_i}{d_{\rm crit}} \right )^{\alpha_2}} + \alpha_{02} \bar {d_i}^{\alpha_{12}} \right )</math>




where <math>\alpha_0</math>, <math>\alpha_1</math>, <math>\alpha_{2}</math>,<math>\alpha_{02}</math>, <math>\alpha_{12}</math> and <math>d_{crit}</math> are coefficients that determine the shape of the energy-specific selection function curve.
where <math>\alpha_0</math>, <math>\alpha_1</math>, <math>\alpha_{2}</math>,<math>\alpha_{02}</math>, <math>\alpha_{12}</math> and <math>d_{\rm crit}</math> are coefficients that determine the shape of the energy-specific selection function curve.


The expanded function reduces to the standard Austin formulation when <math>\alpha_{02}</math> is set to zero.{{King (2012)}}
The expanded function reduces to the standard Austin formulation when <math>\alpha_{02}</math> is set to zero.{{King (2012)}}
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==== Natural selection ====
==== Natural selection ====


Vien (2019) proposed the ''natural selection function'', where the breakage function is dependant on the selection function:{{Vien (2019)}}
Vien (2019) proposed the ''natural selection function'', where the breakage function is dependent on the selection function:{{Vien (2019)}}


:<math>B_{ij} = \dfrac{S^E_i}{S^E_j} \implies b_{ij} = \dfrac{S^E_{i-1} - S^E_i}{S^E_j}</math>
:<math>B_{ij} = \dfrac{S^{\rm E}_i}{S^{\rm E}_j} \implies b_{ij} = \dfrac{S^{\rm E}_{i-1} - S^{\rm E}_i}{S^{\rm E}_j}</math>


The natural selection function therefore eliminates a number of parameters from the Herbst-Fuerstenau model formulation.
The natural selection function therefore eliminates a number of parameters from the Herbst-Fuerstenau model formulation.
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This may assist in the application of additional functional forms or the analysis of experimental data, particularly where the breakage function is non-normalisable.
This may assist in the application of additional functional forms or the analysis of experimental data, particularly where the breakage function is non-normalisable.
</hide>


== Additional notes ==
== Additional notes ==
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Parameters of the Herbst-Fuerstenau model may be adjusted to simulate batch laboratory mills, and continuous rod, ball and stirred mills.
Parameters of the Herbst-Fuerstenau model may be adjusted to simulate batch laboratory mills, and continuous rod, ball and stirred mills.


'''Laboratory mills''' are frequently operated in batch mode. They may be simulated by applying the ''batch'' form of the population balance equations (above) and reformulating <math>\bar E</math> as:
'''Laboratory mills''' are typically operated in batch mode. They may be simulated by applying the ''batch'' form of the population balance equations (above) and reformulating <math>\bar E</math> as:


:<math>\bar E = \dfrac{P.t}{M}</math>
:<math>\bar E = \dfrac{P.t}{M}</math>
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MillFeed= \begin{bmatrix}
MillFeed= \begin{bmatrix}
(Q_m^F)_{11}\text{ (t/h)} & \dots & (Q_m^F)_{1m}\text{ (t/h)}\\  
(Q_{\rm M,F})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,F})_{1m}\text{ (t/h)}\\  
\vdots & \ddots & \vdots\\  
\vdots & \ddots & \vdots\\  
(Q_m^F)_{n1}\text{ (t/h)} & \dots & (Q_m^F)_{nm}\text{ (t/h)}\\  
(Q_{\rm M,F})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,F})_{nm}\text{ (t/h)}\\  
\end{bmatrix}</math>
\end{bmatrix}</math>




:<math>
{{Excel (Text, Inputs, Herbst-Fuerstenau, Selection and Breakage)}}
Selection =
\begin{cases}
 
\begin{bmatrix}
\begin{bmatrix}
S^E_{1}\\
\vdots\\
S^E_{n}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
S^E_{1}\\
\vdots\\
S^E_{n}\\
\end{bmatrix}_m
\end{bmatrix} & \text{ Selection function method}=0\text{ (User defined)}\\
 
& &\\
 
\begin{bmatrix}
\begin{bmatrix}
S^E_{1}\\
\zeta_1\\
\zeta_2\\
d_{p1}\text{ (mm)}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
S^E_{1}\\
\zeta_1\\
\zeta_2\\
d_{p1}\text{ (mm)}\\
\end{bmatrix}_m
\end{bmatrix} & \text{ Selection function method}=1\text{ (Herbst-Fuerstenau)}\\
 
& &\\
 
\begin{bmatrix}
\begin{bmatrix}
\alpha_0\\
\alpha_1\\
\alpha_2\\
d_{crit}\text{ (mm)}\\
\alpha_{02}\\
\alpha_{12}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
\alpha_0\\
\alpha_1\\
\alpha_2\\
d_{crit}\text{ (mm)}\\
\alpha_{02}\\
\alpha_{12}\\
\end{bmatrix}_m
\end{bmatrix} & \text{ Selection function method}=2\text{ (Austin)}\\
 
\end{cases}
</math>
 
 
:<math>
Breakage =
\begin{cases}
 
\begin{bmatrix}
\begin{bmatrix}
b_{11} & & 0\\
\vdots & \ddots & \\
b_{n1} & \dots & b_{nn}\\
\end{bmatrix}_1\\
\vdots\\
\begin{bmatrix}
b_{11} & & 0\\
\vdots & \ddots & \\
b_{n1} & \dots & b_{nn}\\
\end{bmatrix}_m\\
\end{bmatrix} & \text{ Breakage function method}=0\text{ (User defined)}\\
 
& &\\
 
\begin{bmatrix}
\begin{bmatrix}
\beta_0\\
\beta_1\\
\beta_2\\
\beta_{02}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
\beta_0\\
\beta_1\\
\beta_2\\
\beta_{02}\\
\end{bmatrix}_m
\end{bmatrix} & \text{ Breakage function method}=1\text{ (Austin and Luckie)}\\
 
& &\\
 
\begin{bmatrix}
\begin{bmatrix}
K\\
n_1\\
n_2\\
n_3\\
y_0\text{ (mm)}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
K\\
n_1\\
n_2\\
n_3\\
y_0\text{ (mm)}\\
\end{bmatrix}_m
\end{bmatrix} & \text{ Breakage function method}=2\text{ (King)}\\
 
& &\\
 
\bigg [ \;\;\; \bigg ] & \text{ Breakage function method}=3\text{ (Natural selection)}\\


\end{cases}
</math>




where:
where:
* <math>\text{Mixing method}</math> is the population balance equation form, ''0 = Batch/Plug Flow, 1 = Continuous''
* <math>\text{Mixing method}</math> is the population balance equation form, ''0 = Batch/Plug Flow, 1 = Continuous''
* <math>\text{Selection method}</math> is the selection function method, 0 = User, 1 = Herbst-Fuerstenau, 2 = Austin
* <math>\text{Selection method}</math> is the selection function method, ''0 = User, 1 = Herbst-Fuerstenau, 2 = Austin''
* <math>\text{Breakage method}</math> is the breakage function method, ''0 = User, 1 = Austin & Luckie, 2 = King, 3 = Natural selection''
* <math>\text{Breakage method}</math> is the breakage function method, ''0 = User, 1 = Austin & Luckie, 2 = King, 3 = Natural selection''
* <math>m</math> is the number of ore types
* <math>m</math> is the number of ore types
* <math>Q_m^F</math> is feed solids mass flow rate by size and ore type (t/h)
* <math>Q_{\rm M,F}</math> is feed solids mass flow rate by size and ore type (t/h)


=== Results ===
=== Results ===
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\begin{array}{c}
\begin{array}{c}
\begin{bmatrix}
\bar E \text{ (kWh/t)}\\
\bar E \text{ (kWh/t)}\\
&\\
\end{bmatrix}
&\\
\\
\\
\\
\end{array}
\end{array}


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\begin{bmatrix}
\begin{bmatrix}
(Q_m^P)_{11}\text{ (t/h)} & \dots & (Q_m^P)_{1m}\text{ (t/h)}\\
(Q_{\rm M,P})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,P})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
\vdots & \ddots & \vdots\\
(Q_m^P)_{n1}\text{ (t/h)} & \dots & (Q_m^P)_{nm}\text{ (t/h)}\\
(Q_{\rm M,P})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,P})_{nm}\text{ (t/h)}\\
\end{bmatrix}
\end{bmatrix}


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\begin{bmatrix}
\begin{bmatrix}
\begin{bmatrix}
\begin{bmatrix}
S^E_1\\
S^{\rm E}_1\\
\vdots\\
\vdots\\
S^E_n\\
S^{\rm E}_n\\
\end{bmatrix}_1
\end{bmatrix}_1
\dots
\dots
\begin{bmatrix}
\begin{bmatrix}
S^E_1\\
S^{\rm E}_1\\
\vdots\\
\vdots\\
S^E_n\\
S^{\rm E}_n\\
\end{bmatrix}_m
\end{bmatrix}_m
\end{bmatrix}
\end{bmatrix}
Line 427: Line 312:
</math>
</math>


where <math>Q_m^P</math> is product mass flow rate (t/h).
where <math>Q_{\rm M,P}</math> is product mass flow rate (t/h).


=== Example ===
=== Example ===
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The sections and variable names used in the SysCAD interface are described in detail in the following tables.
The sections and variable names used in the SysCAD interface are described in detail in the following tables.


{{SysCAD (Page, Ball Mill, ScdMD*BallMill)|method=1}}
{{SysCAD (Page, Mill, DLL*Mill)|PowerModels=true|MediaTraj=true|MediaStrings=true|Overfilling=true}}


==== Mill page ====
==== Mill page ====


The Mill page is used to specify the input parameters for the mill model.
The Mill page is used to specify the input parameters for the mill model.
{{SysCAD_Table_Header}}
{{SysCAD (Text, Table Header)}}


|-
|-
! colspan="3" style="text-align:left;" |''Requirements''
! colspan="3" style="text-align:left;" |''HerbstFuerstenau''
|-
 
|NumParallelUnits
{{SysCAD (Text, Help Link)}}
|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.
|-
|-
! colspan="3" style="text-align:left;" |''Power''
! colspan="3" style="text-align:left;" |''Power''
|- style="vertical-align:top;"
|- style="vertical-align:top;"
|rowspan="6" | Method
|rowspan="7" | Method
|User
|User
|The user specifies the power input to grinding.
|The user specifies the power input to grinding.
|-
|-
|Hogg and Fuerstenau
|HoggFuerstenau
|The [[Tumbling Mill (Power, Hogg and Fuerstenau)|Hogg and Fuerstenau]] model is used to estimate the the power input to grinding.
|The power input to grinding is the [[Tumbling Mill (Power, Hogg and Fuerstenau)|Hogg and Fuerstenau]] model ''BallsPower'' value.
|-
|-
|MorrellE.GrossPower.Grate
|MorrellE (Grate)
|The [[Tumbling Mill (Power, Morrell Empirical)|Morrell Empirical]] model is used to estimate the the power input to grinding, assuming ''grate'' discharge configuration.
|The power input to grinding is the [[Tumbling Mill (Power, Morrell Empirical)|Morrell Empirical]] model ''NetPower.Grate'' value.
|-
|-
|MorrellE.GrossPower.Overflow
|MorrellE (Overflow)
|The [[Tumbling Mill (Power, Morrell Empirical)|Morrell Empirical]] model is used to estimate the the power input to grinding, assuming ''overflow'' discharge configuration.
|The power input to grinding is the [[Tumbling Mill (Power, Morrell Empirical)|Morrell Empirical]] model ''NetPower.Overflow'' value.
|-
|-
|MorrellC
|MorrellC
|The [[Tumbling Mill (Power, Morrell Continuum)|Morrell Continuum]] model is used to estimate the the power input to grinding.
|The power input to grinding is the [[Tumbling Mill (Power, Morrell Continuum)|Morrell Continuum]] model ''NetPower'' value.
|-
|-
|MorrellD
|MorrellD
|The [[Tumbling Mill (Power, Morrell Discrete Shell)|Morrell Discrete Shell]] model is used to estimate the the power input to grinding.
|The power input to grinding is the [[Tumbling Mill (Power, Morrell Discrete Shell)|Morrell Discrete Shell]] model ''NetPower'' value.
|-
|HildenPowell
|The power input to grinding is the [[Tumbling Mill (Power, Hilden and Powell)|Hilden and Powell]] model ''NetPower'' value.
|-
|-
|MediaPower
|MediaPower
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|Input
|Input
|The number of mixers-in-series for grinding simulation.
|The number of mixers-in-series for grinding simulation.
|-
 
! colspan="3" style="text-align:left;" |''Selection''
{{SysCAD (Text, Mill, Herbst-Fuerstenau, Selection-Breakage)}}
|- style="vertical-align:top;"
|rowspan="3" | Method
|User
|The user specifies the selection function.
|-
|Herbst and Fuerstenau
|The Herbst and Fuerstenau selection function is used.
|-
|Austin
|The Austin selection function is used.
|-
|OreSpecific
|CheckBox
|Ore-specific parameters, allows the selection function to be separately input for all species. Default is all species have the same set of single input properties.
|-
| colspan="3" style="text-align:left;" style="background: #eaecf0" | The fields below are only visible if '''Herbst-Fuerstenau''' is selected.
|-
|S1E
|Input
|Input parameter of the Herbst-Fuerstenau selection function.
|-
|Zeta1
|Input
|Input parameter of the Herbst-Fuerstenau selection function.
|-
|Zeta2
|Input
|Input parameter of the Herbst-Fuerstenau selection function.
|-
|dp1
|Input
|Input parameter of the Herbst-Fuerstenau selection function.
|-
|colspan="3" |
[[File:MillHerbstFuerstenau5.png]] [[File:MillHerbstFuerstenau6.png]]
|-
| colspan="3" style="text-align:left;" style="background: #eaecf0" | The fields below are only visible if '''Austin''' is selected.
|-
|Alpha0
|Input
|Input parameter of the Austin selection function.
|-
|Alpha1
|Input
|Input parameter of the Austin selection function.
|-
|Alpha2
|Input
|Input parameter of the Austin selection function.
|-
|dCrit
|Input
|Input parameter of the Austin selection function.
|-
|Alpha02
|Input
|Input parameter of the Austin selection function.
|-
|Alpha12
|Input
|Input parameter of the Austin selection function.
|-
|colspan="3" |
[[File:MillHerbstFuerstenau7.png]] [[File:MillHerbstFuerstenau8.png]]
|-
! colspan="3" style="text-align:left;" |''Breakage''
|- style="vertical-align:top;"
|rowspan="4" | Method
|User
|The user specifies the breakage function.
|-
|Austin and Luckie
|The Austin and Luckie breakage function is used.
|-
|King
|The King selection function is used.
|-
|Natural Selection
|The Natural selection function is used.
|-
|OreSpecific
|CheckBox
|Ore-specific parameters, allows the selection function to be separately input for all species. Default is all species have the same set of single input properties.
|-
| colspan="3" style="text-align:left;" style="background: #eaecf0" | The fields below are only visible if '''Austin and Luckie''' is selected.
|-
|Beta0
|Input
|Input parameter of the Austin and Luckie breakage function.
|-
|Beta1
|Input
|Input parameter of the Austin and Luckie breakage function.
|-
|Beta2
|Input
|Input parameter of the Austin and Luckie breakage function.
|-
|Beta01
|Input
|Input parameter of the Austin and Luckie breakage function.
|-
|colspan="3" |
[[File:MillHerbstFuerstenau9.png]] [[File:MillHerbstFuerstenau10.png]]
|-
| colspan="3" style="text-align:left;" style="background: #eaecf0" | The fields below are only visible if '''King''' is selected.
|-
|K
|Input
|Input parameter of the King breakage function.
|-
|n1
|Input
|Input parameter of the King breakage function.
|-
|n2
|Input
|Input parameter of the King breakage function.
|-
|n3
|Input
|Input parameter of the King breakage function.
|-
|y0
|Input
|Input parameter of the King breakage function.
|-
|colspan="3" |
[[File:MillHerbstFuerstenau11.png]] [[File:MillHerbstFuerstenau12.png]]


|-
|-
Line 626: Line 384:
|}
|}


==== Selection page ====
{{SysCAD (Page, Mill, Herbst-Fuerstenau, Selection)}}
 
The Selection page is used to specify or display the selection function values.
 
{{SysCAD_Table_Header}}


{{SysCAD_Distribution}}
{{SysCAD (Page, Mill, Herbst-Fuerstenau, Breakage)}}


|-
{{SysCAD (Page, Tumbling Mill, Power)|modelpage={{SysCAD (Text, UnitType Prefix)}}Mill|HF=true|MorrellC=true|MorrellE=true|MorrellD=true|HildenPowell=true|Nitta=true}}
! 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.
|-
|Selection
|style="background: #eaecf0" | Input/Display
| Value of  the selection function for each size interval, for each ore species.
|}
 
==== Breakage page ====
 
The Breakage page is used to specify or display the breakage function values.
 
{{SysCAD_Table_Header}}
 
{{SysCAD_Distribution}}
 
|-
! colspan="3" style="text-align:left;" |''Breakage ''
|-
|Size
|style="background: #eaecf0" | Display
|Size of each interval in internal mesh series.
|-
|Breakage
|style="background: #eaecf0" | Input/Display
| Value of  the breakage function for each parent size interval, progeny size interval and ore species.
|}


{{SysCAD (Page, Tumbling Mill, Power)|modelpage=ScdMD*BallMill|method=0}}
{{SysCAD (Page, Tumbling Mill, MediaStrings)|modelpage={{SysCAD (Text, UnitType Prefix)}}Mill}}


{{SysCAD (Page, Tumbling Mill, MediaTraj)|modelpage=ScdMD*BallMill}}
{{SysCAD (Page, Tumbling Mill, MediaTraj)|modelpage={{SysCAD (Text, UnitType Prefix)}}Mill}}


{{SysCAD (Page, BallMill Overfilling)|modelpage=ScdMD*BallMill}}
{{SysCAD (Page, Ball Mill, Overfilling)|modelpage={{SysCAD (Text, UnitType Prefix)}}Mill}}


{{SysCAD (Page, About)}}
{{SysCAD (Page, About)}}
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* [[Ball Mill (Perfect Mixing)|Perfect Mixing ball mill model]]
* [[Ball Mill (Perfect Mixing)|Perfect Mixing ball mill model]]
* [[HPGR (Torres)|Torres HPGR model]]
* [[HPGR (Torres)|Torres HPGR model]]
* [[HPGR (Campos)|Campos HPGR model]]


== References ==
== References ==

Latest revision as of 10:01, 4 December 2024

Description

This article describes an implementation of the Herbst and Fuerstenau (1980) grinding model.[1]

This model is useful for simulating batch, ball, rod and stirred mill configurations, and is a grinding power-based alternative to the (otherwise very similar) Perfect Mixing model approach.

The model is particularly flexible and offers a number of different equation options for representing the parameters of grinding.

Model theory

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Additional notes

Parameters of the Herbst-Fuerstenau model may be adjusted to simulate batch laboratory mills, and continuous rod, ball and stirred mills.

Laboratory mills are typically operated in batch mode. They may be simulated by applying the batch form of the population balance equations (above) and reformulating as:

where is the batch grinding time (s) and is the total mass of solids in the batch.

Rod mills are often assumed to operate as plug flow devices and may also be simulated by applying the batch form of the population balance equations (above).[2] No reformulation of is required in this case.

Ball mills and stirred mills are frequently assumed to be well-mixed and may be simulated by applying the continuous form of the population balance equations and specifying the number of mixers-in-series, , where appropriate.[2][3]

Excel

The Herbst-Fuerstenau mill model may be invoked from the Excel formula bar with the following function call: 

=mdUnit_Mill_HerbstFuerstenau(Parameters as Range, Size as Range, MillFeed as Range, Selection as Range, Breakage 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 population balance equation form, 0 = Batch/Plug Flow, 1 = Continuous
  • is the selection function method, 0 = User, 1 = Herbst-Fuerstenau, 2 = Austin
  • is the breakage function method, 0 = User, 1 = Austin & Luckie, 2 = King, 3 = Natural selection
  • is the number of ore types
  • is feed solids mass flow rate by size and ore type (t/h)

Results

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

where is product mass flow rate (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 2. Example showing the selection of the Size (red frame) and MillFeed (purple frame) arrays in Excel.
Figure 3. Example showing the selection of the Selection (green frame) and Breakage (pink frame) arrays in Excel.
Figure 4. Example showing the outline of the Results (light blue frame) array 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.
OverfillingIndicator CheckBox Show overflow ball mill slurry volume, residence time, and overfilling evaluation on Overfilling 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
HerbstFuerstenau
HelpLink ButtonModelHelp.png Opens a link to this page using the system default web browser. Note: Internet access is required.
Power
Method User The user specifies the power input to grinding.
HoggFuerstenau The power input to grinding is the Hogg and Fuerstenau model BallsPower value.
MorrellE (Grate) The power input to grinding is the Morrell Empirical model NetPower.Grate value.
MorrellE (Overflow) The power input to grinding is the Morrell Empirical model NetPower.Overflow value.
MorrellC The power input to grinding is the Morrell Continuum model NetPower value.
MorrellD The power input to grinding is the Morrell Discrete Shell model NetPower value.
HildenPowell The power input to grinding is the Hilden and Powell model NetPower value.
MediaPower Input/Display Power input to grinding. The value may be user specified or linked to the selected power model on the Power page.
Mixing
NMixers Input The number of mixers-in-series for grinding simulation.
Selection
Method User The user specifies the selection function.
Herbst and Fuerstenau The Herbst and Fuerstenau selection function is used.
Austin The Austin selection function is used.
OreSpecific CheckBox Ore-specific parameters, allows the selection function 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.
The fields below are only visible if Herbst-Fuerstenau is selected.
S1E Input Input parameter of the Herbst-Fuerstenau selection function.
Zeta1 Input Input parameter of the Herbst-Fuerstenau selection function.
Zeta2 Input Input parameter of the Herbst-Fuerstenau selection function.
dp1 Input Input parameter of the Herbst-Fuerstenau selection function.

MillHerbstFuerstenau5.png MillHerbstFuerstenau6.png

The fields below are only visible if Austin is selected.
Alpha0 Input Input parameter of the Austin selection function.
Alpha1 Input Input parameter of the Austin selection function.
Alpha2 Input Input parameter of the Austin selection function.
dCrit Input Input parameter of the Austin selection function.
Alpha02 Input Input parameter of the Austin selection function.
Alpha12 Input Input parameter of the Austin selection function.

MillHerbstFuerstenau7.png MillHerbstFuerstenau8.png

Breakage
Method User The user specifies the breakage function.
Austin and Luckie The Austin and Luckie breakage function is used.
King The King selection function is used.
Natural Selection The Natural selection function is used.
OreSpecific CheckBox Ore-specific parameters, allows the breakage function 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.
The fields below are only visible if Austin and Luckie is selected.
Beta0 Input Input parameter of the Austin and Luckie breakage function.
Beta1 Input Input parameter of the Austin and Luckie breakage function.
Beta2 Input Input parameter of the Austin and Luckie breakage function.
Beta01 Input Input parameter of the Austin and Luckie breakage function.

MillHerbstFuerstenau9.png MillHerbstFuerstenau10.png

The fields below are only visible if King is selected.
K Input Input parameter of the King breakage function.
n1 Input Input parameter of the King breakage function.
n2 Input Input parameter of the King breakage function.
n3 Input Input parameter of the King breakage function.
y0 Input Input parameter of the King breakage function.

MillHerbstFuerstenau11.png MillHerbstFuerstenau12.png

Results
Ebar Display Specific energy of grinding.

Selection page

The Selection page is used to specify or display the selection function values.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Distribution
Name Display Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream.
IntervalCount Display Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream.
SpWithPSDCount Display Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
Selection
Size Display Size of each interval in mesh series.
MeanSize Display Geometric mean size of each interval in mesh series.
Selection Input/Display Value of the selection function for each size interval, for each ore species.

Breakage page

The Breakage page is used to specify or display the breakage function values.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Distribution
Name Display Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream.
IntervalCount Display Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream.
SpWithPSDCount Display Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
Breakage
Size Display Size of each interval in internal mesh series.
Breakage Input/Display Value of the breakage function for each parent size interval, progeny size interval and 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.
MorrellD CheckBox Shows inputs and results for tumbling mill power calculations using the Morrell Discrete Shell method.
HildenPowell CheckBox Shows inputs and results for tumbling mill power calculations using the Hilden and Powell method.
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.

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.

Overfilling page

This page displays the inputs and results for overflow discharge mill overfilling calculations. The page is only visible if OverfillingIndicator 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.

See also

References

  1. ↑ Herbst, J.A. and Fuerstenau, D.W., 1980. Scale-up procedure for continuous grinding mill design using population balance models. International Journal of Mineral Processing, 7(1), pp.1-31.
  2. ↑ 2.0 2.1 King, R.P., 2012. Modeling and Simulation of Mineral Processing Systems. Elsevier.
  3. ↑ Mazzinghy, D.B., Schneider, C.L., Alves, V.K. and Galéry, R., 2015. Vertical agitated media mill scale-up and simulation. Minerals Engineering, 73, pp.69-76.
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