Ball Mill (Perfect Mixing): Difference between revisions

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== Description ==
== Description ==


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


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


== Model theory ==
== Model theory ==
{{Restricted content}}


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:
<hide>
 
{{Model theory (Text, Mill, Perfect Mixing, Population Balance)}}
::::{|
| 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 ===
=== Discharge rate ===
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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:
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>
:<math>\left(\frac{R}{D^*}\right)_{\rm Sim} = \left(\frac{R}{D^*}\right)_{\rm Orig} \cdot f_{\rm D} \cdot f_{\rm LF} \cdot f_{\rm CS} \cdot f_{\rm WI} \cdot f_{\rm 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>
{{Model theory (Text, Ball Mill, Perfect Mixing, Breakage Scaling)}}


:<math>Factor_{WI}=\left(\frac{WI_{Orig}}{WI_{Sim}}\right)^{0.8}</math>
The grouping of breakage rate scaling factors above, excluding work index, essentially represents a relationship with mill power draw, as noted by Napier-Munn et al., King and others.{{Napier-Munn_et_al. (1996)}}{{King (2012)}}
 
:<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 ===
=== Appearance function ===


The Appearance function describes the mass-by-size distribution of progeny particles resulting from the breakage of parent particles.
{{Model theory (Text, Ball Mill, Perfect Mixing, Appearance)}}
 
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 ===
=== 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>
{{Model theory (Text, Ball Mill, Perfect Mixing, Internal mesh)|Perfect Mixing}}
 
when the intervals are so spaced.


=== Multi-component modelling ===
=== Multi-component modelling ===


The original Perfect Mixing model formulation only considered the properties of a single ore type.
{{Model theory (Text, Ball Mill, Perfect Mixing, Multi-component)|Perfect Mixing|breakage rate}}
 
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 ===
=== Power draw ===
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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)}}
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)}}
</hide>


== Excel ==
== Excel ==
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The Perfect Mixing ball mill model may be invoked from the Excel formula bar with the following function call:
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>
<syntaxhighlight lang="vb">=mdUnit_BallMill_PerfectMixing(Parameters as Range, Size as Range, MillFeed as Range, OreSG As Range, Appearance as Range, WorkIndexSim As Range, R/D*KnotPositions as Range, R/D*KnotsOrig as Range)</syntaxhighlight>


{{Excel (Text, Help, No Arguments)}}
{{Excel (Text, Help, No Arguments)}}
Line 141: Line 76:
:<math>Parameters=
:<math>Parameters=
\begin{bmatrix}
\begin{bmatrix}
D_{Orig}\text{ (m)}\\
D_{\rm Orig}\text{ (m)}\\
LF_{Orig}\text{ (v/v)}\\
{\rm LF}_{\rm Orig}\text{ (v/v)}\\
FracCS_{Orig}\text{ (frac)}\\
(C_{\rm s})_{\rm Orig}\text{ (frac)}\\
WI_{Orig}\text{ (kWh/t)}\\
{\rm WI}_{\rm Orig}\text{ (kWh/t)}\\
Db_{Orig}\text{ (mm)}\\
Db_{\rm Orig}\text{ (mm)}\\
D_{Sim}\text{ (m)}\\
D_{\rm Sim}\text{ (m)}\\
LF_{Sim}\text{ (v/v)}\\
{\rm LF}_{\rm Sim}\text{ (v/v)}\\
FracCS_{Sim}\text{ (frac)}\\
(C_{\rm s})_{\rm Sim}\text{ (frac)}\\
WI_{Sim}\text{ (kWh/t)}\\
{\rm WI}_{\rm Orig}\text{ (kWh/t)}\\
Db_{Sim}\text{ (mm)}\\
Db_{\rm Sim}\text{ (mm)}\\
K\\
K\\
L\text{ (m)}\\
L\text{ (m)}\\
\alpha_{c}\text{ (deg.)}\\
\alpha_{c}\text{ (deg.)}\\
D_t\text{ (m)}\\
D_{\rm t}\text{ (m)}\\
J_B\text{ (v/v)}\\
J_{\rm B}\text{ (v/v)}\\
U\text{ (v/v)}\\
U\text{ (v/v)}\\
\rho_B\text{ (t/m}^{\text{3}}\text{)}\\
\rho_{\rm B}\text{ (t/m}^{\text{3}}\text{)}\\
Q_m^{Liquids}\text{ (t/h)}\\
(Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\
\rho_L\text{ (t/m}^{\text{3}}\text{)}\\
\rho_{\rm L}\text{ (t/m}^{\text{3}}\text{)}\\
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


Line 169: Line 104:


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},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


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


:<math>
:<math>
WI_{Sim}= \begin{bmatrix}
WI_{1}\text{ (kWh/t)} & \dots & WI_m\text{ (kWh/t)}\\
\end{bmatrix},\;\;\;\;\;\;
Appearance= \begin{bmatrix}
Appearance= \begin{bmatrix}
\begin{bmatrix}
\begin{bmatrix}
Line 196: Line 127:
A_{31}\text{ (frac)}\\
A_{31}\text{ (frac)}\\
\end{bmatrix}_m
\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},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


Line 213: Line 148:
where:
where:
* <math>\alpha_{c}</math> is angle between the cone end surface and the vertical direction (degrees)
* <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>D_{\rm 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>J_{\rm B}</math> is the ball charge volume fraction (typically <math>J_{\rm B} = {\rm LF}</math> for ball mills) (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>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>\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_m^{Liquids}</math> is the mass flow feed rate of liquids into the mill (t/h)
* <math>(Q_{\rm M,F})_{\rm L}</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>\rho_{\rm 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>n</math> is the number of size intervals
* <math>m</math> is the number of ore types
* <math>m</math> is the number of ore types
Line 224: Line 159:
* <math>d_i</math> is the size of the square mesh interval that feed mass is retained on (mm)
* <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>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>Q_{\rm 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>\rho_{\rm S}</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>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)
* <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)
Line 238: Line 173:


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


Line 266: Line 201:


\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}


Line 316: Line 251:


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


Line 328: Line 263:


where:
where:
* <math>\text{Mill volumetric feed rate}</math> is the flow rate of pulp into the mill (m<sup>3</sup>/h)
* <math>Q_{\rm V,F}</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>V</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>N</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>\rho_{\rm c}</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{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 (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{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>\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>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>\bar{d}_i</math> is the [[Conversions|geometric mean size]] of the internal mesh series interval that mass is retained on (mm)


Line 353: Line 288:
== SysCAD ==
== SysCAD ==


The SysCAD interface for steady-state (ProBal) mode is described below. For SysCAD Dynamic, see ''[[Ball Mill (Perfect Mixing, Dynamic)]]''.
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=0}}
{{SysCAD (Page, Mill, DLL*Mill)|PowerModels=true|MediaTraj=true|MediaStrings=true|Overfilling=true}}


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


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


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


{{SysCAD (Page, Tumbling Mill, Power)|modelpage=ScdMD*BallMill|method=0}}
{{SysCAD (Page, Tumbling Mill, Power)|modelpage={{SysCAD (Text, UnitType Prefix)}}Mill|HF=true|MorrellC=true|MorrellE=true|MorrellD=true|HildenPowell=true}}
 
{{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)}}

Latest revision as of 10:00, 4 December 2024

Description

This article describes an implementation of the Perfect Mixing ball mill model outlined by Napier-Munn et al. (1996).[1]

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

Model theory

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Excel

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

=mdUnit_BallMill_PerfectMixing(Parameters as Range, Size as Range, MillFeed as Range, OreSG As Range, Appearance as Range, WorkIndexSim As Range, R/D*KnotPositions as Range, R/D*KnotsOrig 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 angle between the cone end surface and the vertical direction (degrees)
  • is the diameter of the discharge trunnion (m)
  • is the ball charge volume fraction (typically for ball mills) (v/v)
  • is the void fill fraction, the volumetric fraction of grinding media interstitial void space occupied by slurry (v/v)
  • is the Specific Gravity or density of the ball media (excluding void space) (- or t/m3)
  • is the mass flow feed rate of liquids into the mill (t/h)
  • is the Specific Gravity or density of liquids in the feed (- or t/m3)
  • is the number of size intervals
  • is the number of ore types
  • is the number of breakage rate per discharge rate knots
  • is the size of the square mesh interval that feed mass is retained on (mm)
  • , i.e. descending size order from top size () to sub mesh ( mm)
  • is the mass flow rate of particles in the feed (t/h)
  • is the Specific Gravity or density of solids (- or t/m3)
  • is the Appearance function value, the fraction of parent particle mass appearing in internal size interval (frac)
  • is breakage rate per discharge rate normalised for residence time (h-1/h-1/h)

Results

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

where:

  • is the flow rate of pulp into the mill (m3/h)
  • is the total volume inside the mill, calculated as the sum of a cylinder and two frustums (m3)
  • is the rotational rate of the mill (rpm)
  • is the combined density of the charge, including grinding media, coarse ore, slurry and void space (t/m3)
  • is the power input to the motor when the mill is rotating but empty (no balls, rocks or slurry) (kW)
  • is the total power input to the motor if the mill is configured with a grate discharge (kW)
  • is the total power input to the motor if the mill is configured with an overflow discharge (kW)
  • is the discharge rate scaling factor
  • is the mass flow rate of particles in the mill product (t/h)
  • is the 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.

Figure 3. Example showing the selection of the Parameters (blue frame) array in Excel.
Figure 4. Example showing the outline of the Results (light blue frame) array in Excel.
Figure 5. Example showing the selection of the Size (red frame), OreSG (green frame) and MillFeed (purple 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 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
PerfectMixing
HelpLink ButtonModelHelp.png Opens a link to this page using the system default web browser. Note: Internet access is required.
Ball
MediaStringsP50 CheckBox
  • Only visible if the MediaStrings option is checked.
  • Replaces the BallSize.Sim user defined value with the MediaSize (All) value from the MediaStrings page.
  • The value of BallSize.Orig should be determined on the same basis for correct scaling of a changed media charge.
Diameter Input The inside liner diameter of the original and simulated ball mills.
BellyLength Input The inside liner belly length of the simulated ball mill, excluding cones.
ConeAngle Input Angle of the feed and discharge end cones, measured as positive displacement from the vertical direction.
TrunnionDiameter Input The inside liner trunnion diameter of the simulated ball mill.
LoadFrac Input The volumetric load fraction of the original and simulated ball mills.
FracCS Input The fraction critical speed of the original and simulated ball mills.
WorkIndex Input Bond Ball Work Index of ore in the original mill.
BallSize Input Characteristic diameter of balls in original and simulated ball mills.
MaxBreakageRateFactor / K Input Parameter relating ball size and the size at which the breakage rate per discharge rate is maximum.
R/DFunction
NumSplineKnots Input Number of spline knots for the function.
Size Input Spline knot size positions.
Ln(R/D*) Input Values of at each spline knot position.
Power
BallVolume Input Volumetric fraction of mill occupied by balls and voids.
VoidFillFraction Input Volumetric fraction of void space between balls occupied by slurry.
BallSG Input Specific Gravity or density of ball media.
Results
MillVolume Display Volume inside mill, including cones.
MillSpeed Display Rotational speed of simulated mill.
ChargeDensity Display Density of charge in simulated mill, including balls, solids and liquids.
NoLoadPower Display Power draw of empty mill (no balls, solids or liquids)
GrossPower.Grate Display Gross power draw of mill in grate configuration
GrossPower.Overflow Display Gross power draw of mill in overflow configuration

Ore page

This page is used to define the comminution properties of SysCAD species with the size distribution quality in the project.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Appearance
DefaultAppearance

ButtonSetAll.png

Sets all species to the the default Broadbent-Callcott Appearance function.
OreSpecific CheckBox
  • Ore-specific parameters, allows the Appearance data 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.
Appearance Input User-specified Appearance function data for all species with size distribution property.

BallMillPerfectMixing6.png BallMillPerfectMixing7.png

WorkIndex
WorkIndex.Sim Input Bond Ball Work Index data for all species with size distribution property.

Ri/Di page

This page displays the scaling factors and breakage rate per discharge rate for each size interval computed by the Perfect Mixing model.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Scaling
RDStar/RD Display Value of the factor for discharge rate scaling.
Diameter Display Value of the mill diameter factor for rate scaling.
LoadFraction Display Value of the load fraction factor for rate scaling.
FracCS Display Value of the fraction critical speed factor for rate scaling.
WorkIndex Display Value of the Work Index factor of each ore species for rate scaling.
Ri/DiStar
Size Display Size of each interval in internal mesh series.
MeanSize Display Geometric mean size of each interval in internal mesh series.
Ri/DiStar Display Value of normalised rate for each size interval, for each ore species.
Ri/Di
Size Display Size of each interval in internal mesh series.
MeanSize Display Geometric mean size of each interval in internal mesh series.
Ri/Di Display Value of rate for each size interval, for each 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.

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. ↑ Napier-Munn, T.J., Morrell, S., Morrison, R.D. and Kojovic, T., 1996. Mineral comminution circuits: their operation and optimisation. Julius Kruttschnitt Mineral Research Centre, Indooroopilly, QLD.
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