View source for Ball Mill (Perfect Mixing, Dynamic)

== Description ==

This article describes a '''''dynamic''''' implementation of the '''Perfect Mixing''' ball 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 ball mill model. For a full description of the steady-state model, see ''[[Ball 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:TumblingMillDimensions2.png|thumb|500px|Figure 2. Schematic of a tumbling mill showing key dimensions.]]

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 [[Ball Mill (Perfect Mixing)#Breakage rate|the steady-state version]].

The steady-state Perfect Mixing model scaling factors for ''mill diameter'', ''load fraction'', ''fraction critical speed'', ''work index'' and ''ball diameter'' are assumed to be explicitly related to the breakage rate and are applied to the <math>R_i</math> function here, i.e.:

:<math>R_{\rm Sim} = R_{\rm Orig} \cdot f_{\rm D} \cdot f_{\rm LF} \cdot f_{\rm CS} \cdot f_{\rm WI} \cdot f_{\rm Db}</math>

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

The [[Ball 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 [[Ball Mill (Perfect Mixing, Dynamic)|Slurry filling and discharge]], below).

=== Discharge rate ===

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

=== Appearance function ===

{{Model theory (Text, Ball 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.

The mill, whether grate of overflow discharge configured, may be in one of a number of slurry filling states at any time:

# Empty (balls only, no slurry)
# Charge void space partially filled with slurry
# Charge void space fully filled with slurry and a slurry pool formed at the toe of the tumbling charge
# Charge void space fully filled, slurry pool is overflowing the trunnion of the mill

The slurry filling states states are illustrated visually in Figure 3 below:

:{|
|+ '''Figure 3. Overflow discharge mill slurry filling states, profile view.'''
|- style="vertical-align:top;"
| [[File:BallMillPerfectMixingDynamic2.png|thumb|325px|1. Empty (balls only, no slurry).]]
| [[File:BallMillPerfectMixingDynamic3.png|thumb|325px|2. Charge void space partially filled with slurry.]]
| [[File:BallMillPerfectMixingDynamic4.png|thumb|325px|3. Charge void space fully filled with slurry and a slurry pool formed at the toe of the tumbling charge.]]
| [[File:BallMillPerfectMixingDynamic5.png|thumb|325px|4. Charge void space fully filled, slurry pool is overflowing the discharge trunnion lip.]]
|}

The filling states are numbered are in progressive order as a mill is filled from an empty state. A mill may transition between the states in forward or backward direction during dynamic simulation as the feed and milling conditions change.

Discharge flow from a mill will commence from different filling states depending on whether the mill is grate or overflow configured.  There is no product from the mill prior to the commencement of discharge flow and hence the discharge rate, <math>D_i</math>, 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>

==== Overflow discharge mill ====

Slurry will not discharge from an overflow mill until filling state 4, once the slurry pool level has risen above the trunnion lip. The volumetric discharge rate of pulp from an overflow mill is then, for practical purposes, equal to the instantaneous volumetric feed rate.

The quantity of slurry in the mill at the point of trunnion overflow is determined by the mill dimensions and charge geometry. The [[Ball Mill (Overfilling)#Shi|method described by Shi (2016)]] is used to compute the volumetric sum of slurry within the charge void space and slurry pool at the overflow condition.{{Shi_(2016)}}

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 an overflow mill.

==== Grate discharge mill ====

Slurry will commence discharging from a grate configured mill from step 2, once the slurry level inside the charge exceeds the level of the outermost apertures in the grate. The discharge rate of a grate mill then increases as the level of slurry held-up in the mill increases.

The [[Tumbling Mill (Slurry Flow)|method of Morrell and Stephenson]] (1996) is used to compute the volumetric discharge flow rate of slurry for mill filling states 2 and 3:{{Morrell_and_Stephenson_(1996)}}

{{Model theory (Text, Slurry Flow, Morrell and Stephenson)}}

The model computes the value of <math>d_{\rm max}</math> to ensure the flow rate of water and solids <math><x_{\rm m}</math> discharged from the mill matches the flow rate required by the Morrell and Stephenson relations at the current slurry hold-up level.

If the mill continues to fill due to excessive feed rate or insufficient discharge capacity, the slurry level will eventually overflow the trunnion lip (filling state 4). Once this occurs, the overflow discharge method described above is used to determine pulp outflow (i.e. <math>d_{\rm max}</math> computed such that mill discharge flow rate = mill feed flow rate).

=== Power draw===

[[File:BallMillPerfectMixingDynamic6.png|thumb|375px|Figure 4. Simplified charge and slurry pool shapes.]]

The dynamic Perfect Mixing ball mill model computes the volume of slurry in the mill at each time step according to the unsteady-state population balance, water hold-up, breakage and discharge relations described above.

Estimation of the power draw of the mill must then take into account the instantaneous volume of slurry that is either within the charge void space or in the pool of excess slurry outside the charge.

The tumbling mill power draw model proposed by [[Tumbling Mill (Power, Hilden and Powell)|Hilden and Powell (2018)]] is based on earlier work by Morrell (1996, 2016) and computes the volume of slurry occupying the four areas illustrated in Figure 4:{{Morrell_(1996a)}}{{Morrell_(2016)}}

:A. In the grinding media interstices above the slurry pool level
:B. In the grinding media interstices below the slurry pool level
:C. In the annular sector at the toe of the charge
:D. In the circular segment settled across the centre of the mill

Slurry in areas A - C contributes to the power draw of the mill. Slurry in area D is evenly distributed on each side of the vertical centreline of the mill and therefore does not contribute to power draw.

This makes the Hilden and Powell model it a convenient formulation for use by the dynamic Perfect Mixing mill model, which is continuously computing the total volume of slurry in the mill during simulation.

== Additional notes ==

The Leung classification function applies for both grate and overflow discharge mills:{{Leung et al. (1987)}}{{Man_(2001)}}
* Grate mills are specified according to the aperture dimensions of installed discharge grates. 
* For overflow mills, the maximum particle size reporting to product may be an appropriate estimate of <math>x_{\rm g}</math> if mill contents are not available.
* Alternatively, overflow mills may be simplified and classification effects ignored by setting <math>C_i=1</math> for all <math>i</math>, if preferred. 
* A value of 1 mm for <math>x_{\rm m}</math> may be an adequate choice for both mill discharge configurations.

== Excel ==

The Perfect Mixing ball 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 ball mill model).

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

<syntaxhighlight lang="vb">=mdUnit_BallMill_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)</syntaxhighlight>

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>Parameters=
\begin{bmatrix}
D_{\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)}\\
Db_{\rm Orig}\text{ (mm)}\\
D_{\rm Sim}\text{ (m)}\\
{\rm LF}_{\rm Sim}\text{ (v/v)}\\
(C_{\rm s})_{\rm Sim}\text{ (frac)}\\
{\rm WI}_{\rm Orig}\text{ (kWh/t)}\\
Db_{\rm Sim}\text{ (mm)}\\
K\\
L\text{ (m)}\\
\alpha_{c}\text{ (deg.)}\\
D_{\rm t}\text{ (m)}\\
J_{\rm B}\text{ (v/v)}\\
U\text{ (v/v)}\\
\rho_{\rm B}\text{ (t/m}^{\text{3}}\text{)}\\
(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}\\
A\text{ (m}^{\text{2}}\text{)}\\
\gamma \text{ (m/m)}\\
r_{\rm n} \text{ (m/m)}\\
k_{\rm m}\\
k_{\rm t}\\
\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},\;\;\;\;\;\;

RKnotPositions= \begin{bmatrix}
d_{1}\text{ (mm)}\\ 
\vdots\\ 
d_{k}\text{ (mm)}\\ 
\end{bmatrix},\;\;\;\;\;\;

RKnotOrig= \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>

where:
* <math>L</math> is the mill (belly) length
* <math>\alpha_{c}</math> is angle between the cone end surface and the vertical direction (degrees)
* <math>D_{\rm t}</math> is the diameter of the discharge trunnion (m)
* <math>J_{\rm B}</math> is the ball charge volume fraction (often <math>J_{\rm B} = {\rm LF}</math>) (v/v)
* <math>U</math> is the void fill fraction, the volumetric fraction of grinding media interstitial void space occupied by slurry (v/v)
* <math>\rho_{\rm B}</math> is the Specific Gravity or density of the ball media (excluding void space) (- or t/m<sup>3</sup>)
* <math>(Q_{\rm M,F})_{\rm L}</math> is the mass flow feed rate of liquids into the mill (t/h)
* <math>\rho_{\rm L}</math> is the Specific Gravity or density of liquids in the feed (- 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 = Grate discharge, 1 = Overflow discharge at trunnion height, 2 = 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}=2</math>) (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>Q_{\rm M,F}</math> is feed mass flow rate (t/h)
* <math>\rho_{\rm S}</math> is Specific Gravity or density (- or t/m<sup>3</sup>)

=== Results ===

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

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

\begin{bmatrix}
Q_{\rm V,F}\text{ (m}^{\text{3}}\text{/h)}\\
V\text{ (m}^{\text{3}}\text{)}\\
N\text{ (rpm)}\\
f_D\text{ (-)}\\
f_{\rm LF}\text{ (-)}\\
f_{\rm CS}\text{ (-)}\\
x_{\rm m(small)}\text{ (mm)}\\
x_{\rm m(large)}\text{ (mm)}\\
\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{)}\\
U\text{ (frac)}\\
(Q_{\rm M,P})_{\rm L}\text{ (t/h)}\\
J_{\rm pg}\text{ (v/v)}\\
J_{\rm po}\text{ (v/v)}\\
J_{\rm max}\text{ (v/v)}\\
J_{\rm p}\text{ (v/v)}\\
J_{\rm pm}\text{ (v/v)}\\
J_{\rm pt}\text{ (v/v)}\\
Q_\text{m}\text{ (m}^{\text{3}}\text{/h)}\\
Q_{\rm t}\text{ (m}^{\text{3}}\text{/h)}\\
Q\text{ (m}^{\text{3}}\text{/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}
(f_{\rm WI})_1 & \dots & (f_{\rm WI})_m
\end{bmatrix}\\

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

\end{array}

\end{bmatrix}
</math>

where:
* <math>Q_{\rm V,f}</math> is the flow rate of pulp into the mill (m<sup>3</sup>/h)
* <math>V</math> is the total volume inside the mill, calculated as the sum of a cylinder and two frustums (m<sup>3</sup>)
* <math>N</math> is the [[Tumbling Mill (Speed)|rotational rate of the mill]] (rpm)
* <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 forming a slurry pool (m<sup>3</sup>)
* <math>\text{Max. slurry in mill (OF)}</math> is the maximum volume of slurry in the mill before trunnion 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>Q_{\rm M,P}</math> is product mass flow rate (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:BallMillPerfectMixingDynamic7.png|left|frame|Figure 5. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]] 
| [[File:BallMillPerfectMixingDynamic8.png|left|frame|Figure 6. 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:BallMillPerfectMixing3.png|left|frame|Figure 7. Example showing the selection of the '''Size''' (red frame), '''OreSG''' (green frame) and '''MillFeed''' (purple frame) arrays in Excel.]] 
| [[File:BallMillPerfectMixingDynamic9.png|left|frame|Figure 8. 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 ''[[Ball Mill (Perfect Mixing)]]''.

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

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

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

{{SysCAD (Page, Ball 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|HildenPowell=true}}

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

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

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

{{SysCAD (Page, About)}}

== See also ==

* [[Ball Mill (Perfect Mixing)|Steady-state Perfect Mixing ball mill model]]
* [[Mill (Herbst-Fuerstenau)| Herbst-Fuerstenau mill model]]

== References ==

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