View source for Mill (Herbst-Fuerstenau)

== Description ==

This article describes an implementation of the '''Herbst and Fuerstenau''' (1980) grinding model.{{Herbst and Fuerstenau (1980)}}

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) [[Ball Mill (Perfect Mixing)|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 ==

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:

::::{|
| style="padding: 10px"|<gallery mode="nolines" widths=950px heights=36px>
File:BallMillPerfectMixing1.png|Figure 1. Schematic diagram of the steady-state population balance adopted by the Herbst-Fuerstenau model.
</gallery>
|}

The steady-state Herbst-Fuerstenau population balance is formulated mathematically as:{{King (2012)}}

:<math>f_i + \sum_{j=1}^{i-1}{b_{ij}S_j \tau p_j} - S_i \tau p_i - p_i = 0</math>

which may be rearranged to yield the mill product in terms of the mill feed and breakage parameters:

:<math>p_i = \dfrac{f_i + \sum_{j=1}^{i-1}{b_{ij}S_j \tau}}{1 + S_i \tau}</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>f_i</math> is the mass fractions 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>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>\tau</math> is the mean residence time of solids in the mill <math>i</math> (s)

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

where:
* <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>W</math> is the mass flow rate of solids through the mill at steady-state (t/h)
* <math>\bar E</math> is the specific energy input to the mill (kWh/t)

The steady-state population balance model is formulated in matrix form for steady-state continuous grinding as:{{Herbst and Fuerstenau (1980)}}

:<math>p = TJT^{-1}f</math>

where:
* <math>f</math> is a vector of the mass fractions of particles in the mill feed  (w/w), <math>f = \{f_i \mid i=1,2,\dots,n \}</math>
* <math>p</math> is a vector of the mass fractions of particles in the mill product  (w/w), <math>p = \{p_i \mid i=1,2,\dots,n \}</math>
* <math>T</math> and <math>J</math> are <math>n</math> ''x'' <math>n</math> dimensional matrices with coefficients determined by the selection (<math>S_i^E</math>) and breakage (<math>b_{ij}</math>) functions

The matrix <math>T</math> is defined as:{{Herbst and Rajamani (1982)}}

:<math>
T_{ij} =
\begin{cases}
0 & i<j\\
\\
1 & 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}
</math>

The matrix <math>J</math> is different for '''batch''', '''plug flow''' or '''continuous''' grinding"{{Herbst and Rajamani (1982)}}

:<math>
J_{ij} =
\begin{cases}
{\rm e}^{-S_i \bar E} & \text{for batch or plug flow 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\\
\end{cases}
</math>

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^{\rm E}_i</math>) and breakage (<math>b_{ij}</math>) functions, which are described below.

=== Energy-specific selection function ===

The following options are available for specifying the energy-specific selection function:
* Herbst and Fuerstenau
* Austin
* User defined

==== Herbst and Fuerstenau ====

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

:<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:
* <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>\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_{{\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 ====

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

:<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_{\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)}}

==== User defined ====

The user may specify the values of the energy-specific selection function at every size interval.

This may assist in the application of additional functional forms or the analysis of experimental data.

=== Breakage function ===

The following options are available for specifying the breakage function:
* Austin and Luckie
* King
* Natural selection
* User defined

==== Austin and Luckie ====

The Austin and Luckie (1972) equation for the non-normalised cumulative breakage function is:{{Austin and Luckie (1972)}}{{Sepulveda (2005)}}

:<math>B_{ij} = \beta_{0j} \left ( \dfrac{d_i}{d_{j+1}} \right )^{\beta_1} + (1 - \beta_{0j}) \left ( \dfrac{d_i}{d_{j+1}} \right )^{\beta_2}</math>

where:

:<math>\beta_{0j}= \beta_{0} {\bar d_i}^{- \beta_{01}}</math>

and:
* <math>B_{ij}</math> is the cumulative breakage function, i.e. <math>b_{ij} = B_{ij} - B_{i+1,j}</math>
* <math>\beta_0</math>, <math>\beta_1</math>, <math>\beta_2</math>, <math>\beta_{0j}</math> and <math>\beta_{01}</math> are coefficients that determine the shape of the cumulative breakage function curve

The Austin and Luckie equation reduces to a normalised breakage function when <math>\beta_{01}</math> is set to zero.

==== King ====

King's equation for the non-normalised cumulative breakage function is is:{{King (2012)}}

:<math>B_{ij} = 
\begin{cases}
K \left( \dfrac{d_i}{y_0} \right )^{n_3} \left ( \dfrac{d_i}{d_j} \right )^{n_1} + (1-K)\left ( \dfrac{d_i}{d_j} \right )^{n_2} & d_i<y_0\\
\\
K \left ( \dfrac{d_i}{d_j} \right )^{n_1} + (1-K)\left ( \dfrac{d_i}{d_j} \right )^{n_2} & d_i \geq y_0\\
\end{cases}
</math>

where <math>K</math>, <math>n_1</math>, <math>n_2</math> and <math>n_3</math> are coefficients that determine the shape of the cumulative breakage function curve, and <math>y_0</math> is the reference parent particle size (usually assumed as 5 mm).

The King equation reduces to a normalised breakage function when <math>y_0</math> is set to zero.

==== Natural selection ====

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

:<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.

==== User defined ====

The user may specify the values of the breakage function at every parent and progeny particle size interval.

This may assist in the application of additional functional forms or the analysis of experimental data, particularly where the breakage function is non-normalisable.

== 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 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:

:<math>\bar E = \dfrac{P.t}{M}</math>

where <math>t</math> is the batch grinding time (s) and <math>M</math> 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).{{King (2012)}} No reformulation of <math>\bar E</math> 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, <math>N</math>, where appropriate.{{King (2012)}}{{Mazzinghy (2014)}}

== Excel ==

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

<syntaxhighlight lang="vb">=mdUnit_Mill_HerbstFuerstenau(Parameters as Range, Size as Range, MillFeed as Range, Selection as Range, Breakage as Range)</syntaxhighlight>

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>Parameters=
\begin{bmatrix}
\text{ Mixing method}\\
\text{ Selection function method}\\
\text{ Breakage function method}\\
P\text{  (kW)}\\
N
\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}</math>


:<math>
Selection = 
\begin{cases}

\begin{bmatrix}
\begin{bmatrix}
S^{\rm E}_{1}\\
\vdots\\
S^{\rm E}_{n}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
S^{\rm E}_{1}\\
\vdots\\
S^{\rm E}_{n}\\
\end{bmatrix}_m
\end{bmatrix} & \text{ Selection function method}=0\text{ (User defined)}\\

& &\\

\begin{bmatrix}
\begin{bmatrix}
S^{\rm E}_{1}\\
\zeta_1\\
\zeta_2\\
d_{{\rm p}1}\text{ (mm)}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
S^{\rm E}_{1}\\
\zeta_1\\
\zeta_2\\
d_{{\rm p}1}\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_{\rm crit}\text{ (mm)}\\
\alpha_{02}\\
\alpha_{12}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
\alpha_0\\
\alpha_1\\
\alpha_2\\
d_{\rm 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:
* <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{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>Q_{\rm M,F}</math> 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:

:<math>
mdUnit\_Mill\_HerbstFuerstenau = 
\begin{bmatrix}

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

&

\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}_{n}\text{ (mm)}\\
\end{bmatrix}

&

\begin{bmatrix}
\begin{bmatrix}
S^{\rm E}_1\\
\vdots\\
S^{\rm E}_n\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
S^{\rm E}_1\\
\vdots\\
S^{\rm E}_n\\
\end{bmatrix}_m
\end{bmatrix}

&

\begin{bmatrix}
\begin{bmatrix}
B_{1,2}\\
\vdots\\
B_{n,2}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
B_{1,2}\\
\vdots\\
B_{n,2}\\
\end{bmatrix}_m
\end{bmatrix}

\end{bmatrix}
</math>

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

=== Example ===

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

{|
|- style="vertical-align:top;"
| [[File:MillHerbstFuerstenau1.png|left|frame|Figure 1. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]|| [[File:MillHerbstFuerstenau2.png|left|frame|Figure 2. Example showing the selection of the '''Size''' (red frame) and '''MillFeed''' (purple frame) arrays in Excel.]] || [[File:MillHerbstFuerstenau3.png|left|frame|Figure 3. Example showing the selection of the '''Selection''' (green frame) and '''Breakage''' (pink frame) arrays in Excel.]] || [[File:MillHerbstFuerstenau4.png|left|frame|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.

{{SysCAD (Page, Mill, DLL*Mill)|PowerModels=true|MediaTraj=true|MediaStrings=true|Overfilling=true}}

==== Mill page ====

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

|-
! colspan="3" style="text-align:left;" |''HerbstFuerstenau''

{{SysCAD (Text, Help Link)}}

|-
! colspan="3" style="text-align:left;" |''Requirements''
{{SysCAD (Text, NumParallelUnits)}}
|-
! colspan="3" style="text-align:left;" |''Power''
|- style="vertical-align:top;"
|rowspan="7" | Method
|User
|The user specifies the power input to grinding.
|-
|HoggFuerstenau
|The power input to grinding is the [[Tumbling Mill (Power, Hogg and Fuerstenau)|Hogg and Fuerstenau]] model ''BallsPower'' value.
|-
|MorrellE (Grate)
|The power input to grinding is the [[Tumbling Mill (Power, Morrell Empirical)|Morrell Empirical]] model ''NetPower.Grate'' value.
|-
|MorrellE (Overflow)
|The power input to grinding is the [[Tumbling Mill (Power, Morrell Empirical)|Morrell Empirical]] model ''NetPower.Overflow'' value.
|-
|MorrellC
|The power input to grinding is the [[Tumbling Mill (Power, Morrell Continuum)|Morrell Continuum]] model ''NetPower'' value.
|-
|MorrellD
|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
|style="background: #eaecf0" | Input/Display
|Power input to grinding. The value may be user specified or linked to the selected power model on the '''Power''' page.
|-
! colspan="3" style="text-align:left;" |''Mixing''
|-
|NMixers
|Input
|The number of mixers-in-series for grinding simulation.

{{SysCAD (Text, Mill, Herbst-Fuerstenau, Selection-Breakage)}}

|-
! colspan="3" style="text-align:left;" |''Results''
|- 
|Ebar
|style="background: #eaecf0" | Display
|Specific energy of grinding.
|}

{{SysCAD (Page, Mill, Herbst-Fuerstenau, Selection)}}

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

{{SysCAD (Page, Tumbling Mill, Power)|modelpage={{Name (Text, Company Name)|nospace=1}}*Mill|HF=true|MorrellC=true|MorrellE=true|MorrellD=true|HildenPowell=true|Nitta=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, Ball Mill, Overfilling)|modelpage={{Name (Text, Company Name)|nospace=1}}*Mill}}

{{SysCAD (Page, About)}}

== See also ==

* [[Ball Mill (Perfect Mixing)|Perfect Mixing ball mill model]]
* [[HPGR (Torres)|Torres HPGR model]]
* [[HPGR (Campos)|Campos HPGR model]]

== References ==

[[Category:Excel]]
[[Category:SysCAD]]