View source for Stirred Mill (Perfect Mixing, Dynamic)

== Description ==

This article describes a '''''dynamic''''' implementation of the '''Perfect Mixing''' vertical stirred mill model outlined by Napier-Munn et al. (1996).{{Napier-Munn et al. (1996)}}

The dynamic version uses the same underlying theory and structure as the steady-state Perfect Mixing stirred mill model. For a full description of the steady-state model, see ''[[Stirred Mill (Perfect Mixing)]]''.

== Model theory ==

{{Model theory (Text, Mill, Perfect Mixing, Population Balance, Dynamic)}}

=== Time step discretisation ===

{{Model theory (Text, Mill, Perfect Mixing, Dynamic, Time Step)}}

=== Breakage rate ===

[[File:StirredMillDimensions1.png|thumb|425px|Figure 2. Schematic of a bottom-fed, gravity-induced vertical stirred mill showing key dimensions. The stirrer shown is a screw type with two starts and three turns per start. (After Napier-Munn et al., 1996).{{Napier-Munn et al. (1996)}}]]

The breakage function, <math>R_i</math>, is an input parameter for the dynamic Perfect Mixing model, replacing the <math>R_i/D_i</math> term appearing in [[Stirred Mill (Perfect Mixing)#Breakage rate|the steady-state version]].

{{Model theory (Text, Stirred Mill, Perfect Mixing, Breakage Proportionality)}}

Adopting the same scaling approach and nomenclature as the [[Ball Mill (Perfect Mixing)|Perfect Mixing ball mill]], the <math>R_i^*</math> rate is scaled by the following relation:

:<math>\left(\frac{R}{D^*}\right)_{\rm Sim} = \left(\frac{R}{D^*}\right)_{\rm Orig} \cdot f_{\rm Ds} \cdot f_{\rm Hb} \cdot f_{\rm Ns} \cdot  f_{\rm S} \cdot  f_{\rm T} \cdot  f_{\rm Db} \cdot f_{\rm WI}</math>

{{Model theory (Text, Stirred Mill, Perfect Mixing, Breakage Scaling)}}

The [[Stirred Mill (Perfect Mixing)#Discharge rate|discharge rate scaling factor]], <math>D^*</math>, is excluded from the breakage rate scaling term and is dealt with separately within the model (see [[Stirred Mill (Perfect Mixing, Dynamic)|Slurry filling and discharge]], below).

=== Discharge rate ===

{{Model theory (Text, Mill, Perfect Mixing, Dynamic, Discharge Rate)}}

Prior work suggests that internal classification within stirred mills may be represented by a [[Partition (Size, Whiten-Beta)|Whiten efficiency curve]] (Hasan et al., 2016).{{Hasan et al. (2016)}} Leung's equation is likely to be an adequate approximation of the same classification behaviour.

=== Appearance function ===

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

=== Internal mesh series ===

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

=== Multi-component modelling ===

{{Model theory (Text, Ball Mill, Perfect Mixing, Multi-component)|Perfect Mixing|breakage rate}}

=== Slurry filling and discharge ===

The unsteady-state population balance and liquid hold-up models described above compute the quantity of slurry in the mill at each discrete time step.

Discharge flow does not commence from an initially empty stirred mill until the available internal volume is filled. The available internal volume consists of the grinding media interstices and any remaining space above the media charge.

The discharge rate, <math>D_i</math>, whilst filling the mill is zero. During this period, the dynamic population balance reduces to a batch mill formulation (with or without feed), as outlined by Whiten (1974):{{Whiten_(1974)}}

:<math>\frac{ds_i(t)}{dt} = f_i + \sum_{j=1}^{i-1}A_{ij}R_js_j - (R_is_i-A_{ii}R_is_i)</math>

Discharge commences once the internal mill volume is filled with slurry. The volumetric discharge rate of pulp from the mill is then, for practical purposes, equal to the instantaneous volumetric feed rate.

The model computes the value of <math>d_{\rm max}</math> to ensure the total flow rate of water plus solids classified for discharge matches the required product pulp outflow rate (i.e. the feed rate) for the mill.

== Excel ==

The Perfect Mixing stirred mill model is not implemented in Excel in dynamic form for practical purposes. Excel is not an ideal platform for dynamic simulation and SysCAD (or similar) is preferred.

The dynamic model is, however, included in Excel in a '''run-to-steady-state''' mode where all feed and input parameters are fixed and time steps are progressed until the computed load and discharge stabilises.

This mode is useful for extracting separated <math>R_i</math> and <math>D_i</math> functions from steady-state data such as plant surveys or other model calibrations (including the steady-state Perfect Mixing stirred mill model).

The ''run-to-steady-state'' dynamic Perfect Mixing stirred mill model may be invoked from the Excel formula bar with the following function call:

<syntaxhighlight lang="vb">=mdUnit_StirredMill_PerfectMixingRiDi(Parameters as Range, Size as Range, MillFeed as Range, OreSG As Range, Appearance as Range, WorkIndexSim as Range, RKnotPositions as Range, RKnotsOrig as Range, Classification as Range, Optional ScalingExponents as Range)</syntaxhighlight>

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>Parameters=
\begin{bmatrix}
D\text{ (m)}\\
H\text{ (m)}\\
(H_{\rm b})_{\rm Orig}\text{ (m)}\\
(D_{\rm b})_{\rm Orig}\text{ (mm)}\\
(D_{\rm s})_{\rm Orig}\text{ (m)}\\
(N_{\rm s})_{\rm Orig}\text{ (rpm)}\\
S_{\rm Orig}\\
T_{\rm Orig}\\
(H_{\rm b})_{\rm Sim}\text{ (m)}\\
(D_{\rm b})_{\rm Sim}\text{ (mm)}\\
(D_{\rm s})_{\rm Sim}\text{ (m)}\\
(N_{\rm s})_{\rm Sim}\text{ (rpm)}\\
S_{\rm Sim}\\
T_{\rm Sim}\\
{\rm WI}_{\rm Orig}\text{ (kWh/t)}\\
(Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\
\rho_{\rm L}\text{ (t/m}^{\text{3}}\text{)}\\
\text{Classification method}\\
x_{\rm m} \text{ (mm)}\\
\text{Discharge type}\\
\text{User overflow volume (m}^{\text{3}}\text{)}\\
\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},\;\;\;\;\;\;

OreSG= \begin{bmatrix}
(\rho_{\rm S})_{1}\text{ (t/m}^\text{3}\text{)} & \dots & (\rho_{\rm S})_m\text{ (t/m}^\text{3}\text{)}\\ 
\end{bmatrix}
</math>

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

R/D^*KnotPositions= \begin{bmatrix}
d_{1}\text{ (mm)}\\ 
\vdots\\ 
d_{k}\text{ (mm)}\\ 
\end{bmatrix},\;\;\;\;\;\;

R/D^*KnotOrig= \begin{bmatrix}
\ln\left(\frac{R}{D^*}\right)_1\\ 
\vdots\\ 
\ln\left(\frac{R}{D^*}\right)_k\\ 
\end{bmatrix}</math>

:<math>Classification =
    \begin{cases}
      \begin{bmatrix}
      C_{1}\text{ (frac)}\\
      \vdots\\
      C_{31}\text{ (frac)}\\
      \end{bmatrix} & \mbox{if Classification method}=0\mbox{ (User)}\\      
      x_{\rm g} & \mbox{if Classification method}=1\mbox{ (Leung)}\\
    \end{cases}
</math>

:<math>
\mathit{ScalingExponents} = \begin{bmatrix}
e_1\\ 
e_2\\ 
e_3\\ 
e_4\\ 
e_5\\ 
e_6\\ 
e_7\\ 
\end{bmatrix}^*
</math>

where:
* <math>(Q_{\rm M,F})_{\rm L}</math> is the mass flow feed rate of liquids into the mill (t/h)
* <math>\rho_{\rm B}</math> is the Specific Gravity or density of the media in the mill (- or t/m<sup>3</sup>)
* <math>\rho_{\rm L}</math> is the Specific Gravity or density of liquids in the feed (- or t/m<sup>3</sup>)
* <math>m</math> is the number of ore types
* <math>k</math> is the number of breakage rate per discharge rate knots
* <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)
* <math>Q_{\rm M,F}</math> is the mass flow rate of particles in the feed (t/h)
* <math>\rho_{\rm S}</math> is the Specific Gravity or density of solids (- or t/m<sup>3</sup>)
* <math>\text{Classification method }</math> is the method used to defined the classification-by-size to discharge, ''0 = User-defined partition'' or ''1 = Leung method''
* <math>\text{Discharge type}</math> is discharge configuration, ''0 = Overflow discharge at mill height, 1 = Overflow discharge at user-defined slurry filling volume''
* <math>\text{User overflow volume}</math> is the user-specified slurry filling volume at which overflow commences (if <math>\text{Discharge type}=1</math>) (m<sup>3</sup>)
* <math>^*</math> indicates the <math>\mathit{ScalingExponents}</math> array is optional, the default values are used if omitted.

=== Results ===

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

:<math>
mdUnit\_StirredMill\_PerfectMixing\_RiDi = 
\begin{bmatrix}

\begin{bmatrix}
Q_{\rm V}\text{ (m}^{\text{3}}\text{/h)}\\
V_{\rm Mill}\text{ (m}^{\text{3}}\text{)}\\
V_{\rm Media}\\
V_{\rm Slurry}\\
{\rm LF}\text{ (v/v)}\\
\tau\text{ (min)}\\
\text{R/D* factor (frac)}\\
f_{\rm Ds}\text{ (-)}\\
f_{\rm Hb}\text{ (-)}\\
f_{\rm Ns}\text{ (-)}\\
f_{\rm S}\text{ (-)}\\
f_{\rm T}\text{ (-)}\\
f_{\rm Db}\text{ (-)}\\
f_{\rm WI}\text{ (-)}\\
\text{Iterations}\\
dt\text{ (s)}\\
d_{\rm max}\text{ (h}^{\text{-1}}\text{)}\\
\text{Ore mass (t)}\\
\text{Liquid mass (t)}\\
\text{Ball mass (t)}\\
\text{Slurry in mill (m}^{\text{3}}\text{)}\\
\text{Max. slurry in charge (m}^{\text{3}}\text{)}\\
\text{Max. slurry in mill (OF) (m}^{\text{3}}\text{)}\\
(Q_{\rm M,P})_{\rm L}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{array}{cccccccc}

\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}
M_{11}\text{ (t/h)} & \dots & M_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
M_{n1}\text{ (t/h)} & \dots & M_{nm}\text{ (t/h)}\\
\end{bmatrix}

&
\begin{bmatrix}
\bar{d}_1\text{ (mm)}\\
\vdots\\
\bar{d}_{31}\text{ (mm)}\\
\end{bmatrix}

&

\begin{bmatrix}
R_{11}\left(\text{h}^\text{-1}\right) & \dots & R_{1m}\left(\text{h}^\text{-1}\right)\\
\vdots & \ddots & \vdots \\
R_{31,1}\left(\text{h}^\text{-1}\right) & \dots & R_{31m}\left(\text{h}^\text{-1}\right)\\
\end{bmatrix}

&

\begin{bmatrix}
D_{1}\left(\text{h}^\text{-1}\right)\\
\vdots\\
D_{31}\left(\text{h}^\text{-1}\right)\\
\end{bmatrix}

&

\begin{bmatrix}
C_{1}\text{ (frac)}\\
\vdots\\
C_{31}\text{ (frac)}\\
\end{bmatrix}

&

\begin{bmatrix}
\ln R_{11} & \dots & \ln R_{1m}\\
\vdots & \ddots & \vdots \\
\ln R_{k1} & \dots & \ln R_{km}\\
\end{bmatrix}

\\
& & & & & & &
\\
& & & & & & &

\begin{bmatrix}
R_{11}\left(\text{h}^\text{-1}\right) & \dots & R_{1m}\left(\text{h}^\text{-1}\right)\\
\vdots & \ddots & \vdots \\
R_{k1}\left(\text{h}^\text{-1}\right) & \dots & R_{km}\left(\text{h}^\text{-1}\right)\\
\end{bmatrix}

\\
& & & & & & &
\\
& & & & & & &
\begin{bmatrix}
(Factor_{\rm WI})_1 & \dots & (Factor_{\rm WI})_m
\end{bmatrix}\\

& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
& & & & & &\\
\end{array}

\end{bmatrix}

</math>

where:
* <math>{\rm LF}</math> is the media load fraction (v/v), defined as the fraction of mill volume occupied by media, including void spaces, i.e. <math>{\rm LF} = H_{\rm b} \big/ H</math>
* <math>\text{R/D* factor}=D_i^*/D_i</math> is the discharge rate scaling factor
* <math>Q_{\rm M,P}</math> is the mass flow rate of particles in the mill product (t/h)
* <math>\bar{d}_i</math> is the [[Conversions|geometric mean size]] of the internal mesh series interval that mass is retained on (mm)
* <math>\text{Iterations}</math> is the number of time steps required to reach steady-state
* <math>dt</math> is the size of the discretised time step calculated by the model, <math>\Delta t</math> (s)
* <math>\text{Ore mass}</math> is the total mass of ore in the mill at steady-state (t)
* <math>\text{Liquid mas}</math> is the mass of liquids in the mill at steady-state (t)
* <math>\text{Ball mass}</math> is the mass of balls in the mill at steady-state (t)
* <math>\text{Slurry in mill}</math> is the volume of slurry in the mill at steady-state (m<sup>3</sup>)
* <math>\text{Max. slurry in charge}</math> is the maximum volume of slurry that can occupy the charge void space before (m<sup>3</sup>)
* <math>\text{Max. slurry in mill (OF)}</math> is the maximum volume of slurry in the mill before overflow commences (m<sup>3</sup>)
* <math>(Q_{\rm M,P})_{\rm L}</math> is the discharge mass flow rate of liquids from the mill (t/h)
* <math>M</math> is the mass of solids in the mill (t)

=== Example ===

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

{| 
|- style="vertical-align:top;"
| [[File:StirredMillPerfectMixingDynamic1.png|left|frame|Figure 4. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]] 
| [[File:StirredMillPerfectMixingDynamic2.png|left|frame|Figure 5. Example showing the selection of the '''Appearance''' (pink frame), '''RKnotPositions''' (teal frame), '''RKnotsOrig''' (blue frame), '''WorkIndexSim''' (dark red frame), '''Classification''' (Leung, <math>x_{\rm g}</math>) (red frame) arrays in Excel.]] 
|}
{| 
|- style="vertical-align:top;"
| [[File:StirredMillPerfectMixingDynamic3.png|left|frame|Figure 6. Example showing the selection of the '''Size''' (red frame), '''OreSG''' (green frame) and '''MillFeed''' (purple frame) arrays in Excel.]] 
| [[File:StirredMillPerfectMixingDynamic4.png|left|frame|Figure 7. Example showing the outline of the '''Results''' (light blue frame) array in Excel.]]
|}

== SysCAD ==

The SysCAD interface for Dynamic mode is described below. For steady-state, see ''[[Stirred Mill (Perfect Mixing)]]''.

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

{{SysCAD (Page, Stirred Mill, Perfect Mixing, Mill)|method=1}}

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

{{SysCAD (Page, Stirred Mill, Perfect Mixing, Ri/Di)|method=1}}

{{SysCAD (Page, Ball Mill, Perfect Mixing, Load)}}

{{SysCAD (Page, Ball Mill, Perfect Mixing, Content)}}

{{SysCAD (Page, Tumbling Mill, Power)|modelpage={{SysCAD (Text, UnitType Prefix)}}Mill|Nitta=true}}

{{SysCAD (Page, Tumbling Mill, MediaStrings)|modelpage={{SysCAD (Text, UnitType Prefix)}}Mill}}

{{SysCAD (Page, About)}}

== See also ==

* [[Stirred Mill (Perfect Mixing)|Steady-state Perfect Mixing stirred mill model]]
* [[Mill (Herbst-Fuerstenau)| Herbst-Fuerstenau mill model]]
* [[Stirred Mill (Power, Nitta)]]
* [[Stirred Mill (Power, Heath)]]

== References ==

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