View source for Ball Mill (Perfect Mixing)

== Description ==

This article describes an implementation of the '''Perfect Mixing''' ball mill model outlined by Napier-Munn et al.{{Napier-Munn_et_al._(1996)}}

The model described here is for '''''steady-state''''' simulation. For dynamic simulation, ''see [[Ball Mill (Perfect Mixing, Dynamic)]]''.

== Model theory ==

The Perfect Mixing 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 Perfect Mixing model.
</gallery>
|}

The steady-state population balance is formulated mathematically as:

:<math>f_i-R_is_i+\sum_{j=1}^{i}{A_{ij}R_js_j} - p_i = 0</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 flow rate of solids of size interval <math>i</math> in the mill feed
* <math>p_i</math> is the mass flow rate of solids of size interval <math>i</math> in the mill product
* <math>s_i</math> is the mass of solids on size interval <math>i</math> in the mill load
* <math>R_i</math> is the breakage rate of solids on size interval <math>i</math> in the mill load 
* <math>A_{ij}</math> is the Appearance function, the distribution of particle mass arising from the breakage of a parent particle in size interval <math>j</math> into progeny of size interval <math>i</math>

As the mill is perfectly mixed, the product is related to the mill contents and discharge rate as:

:<math>p_i=D_is_i</math>

where <math>D_i</math> is the rate of discharge of solids in size interval <math>i</math> from the mill.

Substitution and rearrangement of the above equations leads to:

:<math>p_i=\dfrac{f_i + \sum\limits_{j=1}^{i}{A_{ij}\dfrac{R_j}{D_j}p_j}}{1+\dfrac{R_i}{D_i}}</math>

The mill product can therefore be computed provided a feed rate, Appearance function and ''breakage rate per discharge rate'', <math>R_i/D_i</math>, is available. Alternatively, the <math>R_i/D_i</math> function can be determined from the feed rate, product rate and Appearance function.

=== Discharge rate ===

[[File:TumblingMillDimensions2.png|thumb|500px|Figure 2. Schematic of a tumbling mill showing key dimensions.]]

The discharge rate, <math>D_i</math>, is related to the residence time of pulp in the mill and can be corrected for different volumetric feed rates and mill volumes by first normalising to 

:<math>D_i^* = \dfrac{D_i}{\left(\dfrac{4Q_v}{D^2L}\right)}</math>

where:
* <math>D_i^*</math> is the residence time normalised discharge rate per size interval <math>i</math>
* <math>Q_v</math> is the volumetric flow rate of pulp in mill feed (solids plus liquids)
* <math>D</math> is the mill diameter (inside liners)
* <math>L</math> is the mill (belly) length

The actual discharge rate for a given mill and pulp feed rate is subsequently scaled as 

:<math>D_i=D_i^* \cdot \dfrac{4Q_v}{D^2L}</math>

with <math>R_i/D_i</math> becoming <math>R_i/D_i^*</math> for mill model calibration and simulation.

=== Breakage rate ===

The <math>R_i/D_i^*</math> rate is a function of particle size and is typically a smooth, concave downward curve with a maximum related to ball size. In order to reduce the number of model parameters, the rates are specified only at three or four regularly spaced sizes (''knots''). Cubic spline interpolation is then used to reconstruct the complete set of rates for all size intervals.

The breakage rate, <math>R_i</math>, is affected by mill operating conditions and the <math>R_i/D_i^*</math> rate is further scaled by the following relation:

:<math>\left(\frac{R}{D^*}\right)_{Sim} = \left(\frac{R}{D^*}\right)_{Fit} \cdot Factor_{D} \cdot Factor_{LF} \cdot Factor_{FracCS} \cdot Factor_{WI} \cdot Factor_{Db}</math>

The scaling factors are defined as:

:<math>Factor_{D}=\sqrt{\frac{D_{Sim}}{D_{Orig}}}</math>

:<math>Factor_{LF}=\frac{(1-LF_{Sim}).LF_{Sim}}{(1-LF_{Orig}).LF_{Orig}}</math>

:<math>Factor_{FracCS}=\frac{FracCS_{Sim}}{FracCS_{Orig}}</math>

:<math>Factor_{WI}=\left(\frac{WI_{Orig}}{WI_{Sim}}\right)^{0.8}</math>

:<math> Factor_{Db} =
    \begin{cases}
      \dfrac{Db_{Orig}}{Db_{Sim}} & \text{for }x<x_{m(small)}, \;\;\;x_{m(small)}=\min{\left(K.Db_{Orig}^2, K.Db_{Sim}^2\right)}\\      
      \left(\dfrac{Db_{Orig}}{Db_{Sim}}\right)^2 & \text{for }x\geq x_{m(large)}, \;\;\;x_{m(large)}=\max{\left(K.Db_{Orig}^2, K.Db_{Sim}^2\right)}\\
    \end{cases}
</math>

where:
* <math>D</math> is mill diameter (m)
* <math>LF</math> is load fraction, the load volume as a fraction of mill volume (v/v)
* <math>FracCS</math> is the fraction critical speed of the mill (frac)
* <math>WI</math> is the Bond Ball Work Index of the ore (kWh/t)
* <math>Db</math> is the ball diameter (mm)
* <math>x</math> is the diameter of a particle of size interval <math>i</math> (mm)
* <math>K</math> is the maximum breakage rate factor which relates ball size and the size at which <math>R_i/D_i^*</math> is maximum, i.e. <math>x_m=K.D_b^2</math>
* <math>Factor_{Db}</math> is interpolated for <math>x_{m(small)}<x<x_{m(large)}</math>

and the <math>Orig</math> subscript refers to the original mill from which <math>R_i/D_i^*</math> was derived and <math>Sim</math> refers to the mill being simulated (scaled).

The grouping of breakage rate scaling factors above, excluding work index, essentially represent a relationship with mill power draw, as noted by Napier-Munn et al., King and others.{{Napier-Munn_et_al. (1996)}}{{King (2012)}}

=== Appearance function ===

The Appearance function describes the mass-by-size distribution of progeny particles resulting from the breakage of parent particles.

The Appearance function may be specified for a particular ore. Alternatively, the default Broadbent-Callcott appearance function may be used, which is defined as:{{Gupta_and_Yan_(2016)}}

:<math>A_{ij}=\frac{1-\exp \left(-\dfrac{d_i}{d_j} \right)}{1-\exp(-1)}</math>

where <math>d_i</math> is the breakage product particle size and <math>d_j</math> is the original particle size.

=== Internal mesh series ===
The Perfect Mixing ball mill model is formulated internally with a geometric progression of 31 mesh sizes at <math>\sqrt{2}</math> intervals. Feed and product size fractions are automatically converted to and from the internal mesh series during model computation. The <math>\sqrt{2}</math> size intervals allow the Appearance function to be specified as a one-dimensional matrix, rather than the two dimensional form defined above, since 

:<math>A_{ij}=A_{i-j}</math>

when the intervals are so spaced.

=== Multi-component modelling ===

The original Perfect Mixing model formulation only considered the properties of a single ore type.

This model internally applies different Appearance functions and breakage rate scaling to separate population balance computations for each ore type (mineral) in the feed, as conceptually suggested by Napier-Munn et al.<ref name="Napier-Munn_et_al._(1996)" />

=== Power draw ===

The Perfect Mixing mill model formulation does not explicitly include a relationship with mill power draw, other than the breakage rate scaling observations noted above.

However, a simple estimate of power draw is provided for user convenience. Power is calculated according to the [[Tumbling Mill (Power, Morrell Empirical)|Morrell Empirical]] approach for grate and overflow discharge mills.{{Morrell_(1996b)}}

== Excel ==

The Perfect Mixing ball mill model may be invoked from the Excel formula bar with the following function call:

<syntaxhighlight lang="vb">=mdUnit_BallMill_PerfectMixing(Parameters as Range, Size as Range, MillFeed as Range, OreSG As Range, WorkIndexSim As Range, Appearance as Range, R/D*KnotPositions as Range, R/D*KnotsOrig as Range)</syntaxhighlight>

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>Parameters=
\begin{bmatrix}
D_{Orig}\text{ (m)}\\
LF_{Orig}\text{ (v/v)}\\
FracCS_{Orig}\text{ (frac)}\\
WI_{Orig}\text{ (kWh/t)}\\
Db_{Orig}\text{ (mm)}\\
D_{Sim}\text{ (m)}\\
LF_{Sim}\text{ (v/v)}\\
FracCS_{Sim}\text{ (frac)}\\
WI_{Sim}\text{ (kWh/t)}\\
Db_{Sim}\text{ (mm)}\\
K\\
L\text{ (m)}\\
\alpha_{c}\text{ (deg.)}\\
D_t\text{ (m)}\\
J_B\text{ (v/v)}\\
U\text{ (v/v)}\\
\rho_B\text{ (t/m}^{\text{3}}\text{)}\\
Q_m^{Liquids}\text{ (t/h)}\\
\rho_L\text{ (t/m}^{\text{3}}\text{)}\\
\end{bmatrix},\;\;\;\;\;\;

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

MillFeed= \begin{bmatrix}
(Q_m^F)_{11}\text{ (t/h)} & \dots & (Q_m^F)_{1m}\text{ (t/h)}\\ 
\vdots & \ddots & \vdots\\ 
(Q_m^F)_{n1}\text{ (t/h)} & \dots & (Q_m^F)_{nm}\text{ (t/h)}\\ 
\end{bmatrix},\;\;\;\;\;\;

OreSG= \begin{bmatrix}
SG_{1}\text{ (t/m}^\text{3}\text{)} & \dots & SG_m\text{ (t/m}^\text{3}\text{)}\\ 
\end{bmatrix}
</math>

:<math>
WI_{Sim}= \begin{bmatrix}
WI_{1}\text{ (kWh/t)} & \dots & WI_m\text{ (kWh/t)}\\ 
\end{bmatrix},\;\;\;\;\;\;

Appearance= \begin{bmatrix}
\begin{bmatrix}
A_{1}\text{ (frac)}\\
\vdots\\
A_{31}\text{ (frac)}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
A_{1}\text{ (frac)}\\
\vdots\\
A_{31}\text{ (frac)}\\
\end{bmatrix}_m
\end{bmatrix},\;\;\;\;\;\;

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>

where:
* <math>\alpha_{c}</math> is angle between the cone end surface and the vertical direction (degrees)
* <math>D_t</math> is the diameter of the discharge trunnion (m)
* <math>J_B</math> is the ball charge volume fraction (often <math>J_B = 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_B</math> is the specific gravity or density of the ball media (excluding void space) (- or t/m<sup>3</sup>)
* <math>Q_m^{Liquids}</math> is the mass flow feed rate of liquids into the mill (t/h)
* <math>\rho_L</math> is the specific gravity or density of liquids in the feed (- or t/m<sup>3</sup>)
* <math>n</math> is the number of size intervals
* <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_m^F</math>  is the mass flow rate of particles in the feed (t/h)
* <math>SG</math> is the Specific Gravity or density of solids (- or t/m<sup>3</sup>)
* <math>A_i</math> is the Appearance function value, the fraction of parent particle mass appearing in internal size interval <math>i</math> (frac)
* <math>\frac{R_i}{D_i^*}</math> is breakage rate per discharge rate normalised for residence time (h<sup>-1</sup>/h<sup>-1</sup>/h)

=== Results ===

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

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

\begin{bmatrix}
\text{Mill volumetric feed rate (m}^{\text{3}}\text{/h)}\\
\text{Mill volume (m}^{\text{3}}\text{)}\\
\text{Mill speed (rpm)}\\
\text{Charge density (t/m}^{\text{3}}\text{)}\\
\text{No-load power (kw)}\\
\text{Gross power (grate) (kW)}\\
\text{Gross power (overflow) (kW)}\\ 
\text{R/D* factor (frac)}\\
Factor_D\text{ (frac)}\\
Factor_{LF}\text{ (frac)}\\
Factor_{CS}\text{ (frac)}\\
xm(small)\text{ (mm)}\\
xm(large)\text{ (mm)}\\
\end{bmatrix}

&

\begin{array}{ccc}

\begin{bmatrix}
d_1\text{ (mm)}\\
\vdots\\
d_n\text{ (mm)}
\end{bmatrix}

&

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

&

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

&

\begin{bmatrix}
\frac{R_i}{D_i^*}_{11}\left(\frac{\text{h}^\text{-1}}{\text{h}^\text{-1}}/\text{h}\right) & \dots & \frac{R_i}{D_i^*}_{1m}\left(\frac{\text{h}^\text{-1}}{\text{h}^\text{-1}}/\text{h}\right)\\
\vdots & \ddots & \vdots \\
\frac{R_i}{D_i^*}_{31,1}\left(\frac{\text{h}^\text{-1}}{\text{h}^\text{-1}}/\text{h}\right) & \dots & \frac{R_i}{D_i^*}_{31m}\left(\frac{\text{h}^\text{-1}}{\text{h}^\text{-1}}/\text{h}\right)\\
\end{bmatrix}

&

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

&

\begin{bmatrix}
\ln \left( \frac{R_i}{D_i} \right)_{11} & \dots & \ln \left( \frac{R_i}{D_i} \right)_{1m}\\
\vdots & \ddots & \vdots \\
\ln \left( \frac{R_i}{D_i} \right)_{k1} & \dots & \ln \left( \frac{R_i}{D_i} \right)_{km}\\
\end{bmatrix}\\

\\
& & & & & 

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

\\
& & & & & 

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

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

\end{array}

\end{bmatrix}
</math>

where:
* <math>\text{Mill volumetric feed rate}</math> is the flow rate of pulp into the mill (m<sup>3</sup>/h)
* <math>\text{Mill volume}</math> is the total volume inside the mill, calculated as the sum of a cylinder and two frustums (m<sup>3</sup>)
* <math>\text{Mill speed}</math> is the [[Tumbling Mill (Speed)|rotational rate of the mill]] (rpm)
* <math>\text{Charge density}</math> is the combined density of the charge, including grinding media, coarse ore, slurry and void space (t/m<sup>3</sup>)
* <math>\text{No-load power}</math> is the power input to the motor when the mill is rotating but empty (no balls, rocks or slurry) (kW)
* <math>\text{Gross power (grate)}</math> is the total power input to the motor if the mill is configured with a grate discharge (kW)
* <math>\text{Gross power (overflow)}</math> is the total power input to the motor if the mill is configured with an overflow discharge (kW)
* <math>\text{R/D* factor}=D_i^*/D_i</math> is the discharge rate scaling factor
* <math>Q_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)

=== Example ===

The images below show the selection of input arrays and output results in the Excel interface.
{|
|- style="vertical-align:top;"
| [[File:BallMillPerfectMixing2.png|left|frame|Figure 3. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]

[[File:BallMillPerfectMixing5.png|left|frame|Figure 4. Example showing the outline of the '''Results''' (light blue frame) array in Excel.]]

|| [[File:BallMillPerfectMixing3.png|left|frame|Figure 5. Example showing the selection of the '''Size''' (red frame), '''OreSG''' (green frame) and '''MillFeed''' (purple frame).]] || [[File:BallMillPerfectMixing4.png|left|frame|Figure 6. Example showing the selection of the '''Appearance''' (pink frame), '''R/D*KnotPositions''' (teal frame), '''R/D*KnotsOrig''' (blue frame), '''WorkIndexSim''' (red frame) arrays in Excel.]] 
|}

== SysCAD ==

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

{{SysCAD (Page, Ball Mill, ScdMD*BallMill)|method=0}}

{{SysCAD (Page, Ball Mill, Perfect Mixing, Mill)|method=0}}

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

{{SysCAD (Page, Ball Mill, Perfect Mixing, Ri/Di)|method=0}}

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

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

{{SysCAD (Page, BallMill Overfilling)|modelpage=ScdMD*BallMill}}

{{SysCAD (Page, About)}}

== See also ==

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

== References ==

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