AG/SAG Mill (Variable Rates): Difference between revisions

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{{Under construction|section}}
{{Under construction|section}}
<!--
<!--
Introductory text here.
Algorithm block diagram somewhere here. Calculation sequence. As implemented, especially ball-ore since combined ore SG can change during solve.


=== Perfect mixing model ===
=== Perfect mixing model ===


Perfect mixing model rearranged to solve for Load (si).{{Napier-Munn et al. (1996)}}
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:{{Valery_and_Morrell_(1995)}}
 
:<math>f_i + \sum_{j=1}^{i-1}A_{ij}R_js_j - (R_is_i-A_{ii}R_is_i) - 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 of size interval <math>i</math> in the mill load
* <math>R_i</math> is the breakage rate of solids of 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.
 
Therefore, the mill load at steady-state can be computed from:
 
:<math>s_i= \dfrac{f_i + \sum\limits_{j=1}^{i-1}{A_{ij} R_j s_j}}{D_i + R_i - A_{ii} R_i}</math>
 
and the product <math>p_i</math> subsequently determined.
 
Liquids retained in the mill at steady-state are similarly determined from:
 
:<math>f_{\rm w} - D_{\rm w} s_{\rm w} = 0</math>
 
where:
* <math>s_{\rm w}</math> is the load mass of water in the mill
* <math>f_{\rm w}</math> is the mass feed rate of water into the mill
* <math>D_{\rm w}</math> is the discharge rate of water from the mill, normally assumed to equal the value of <math>D_i</math> at the finest size interval.


=== Slurry flow ===
=== Slurry flow ===


1999, T.J. Napier-Munn - S. Morrell, Mineral Comminution Circuits - Their Operation and Optimisation.{{Napier-Munn et al. (1996)}}
The volumetric flow rate of slurry discharged from a grated mill is a function of the slurry hold-up within the mill, in a manner analogous to flow from the bottom of a filled tank.
 
In a steady-state model, the discharge flow rate is equal to the feed flow rate by definition. The following empirical relationship is used to estimate slurry hold-up in a grated mill for a given discharge flow rate:{{Napier-Munn et al. (1996)}}


1996, Morrell, Stephenson, Slurry discharge capacity of autogenous and semi - autogenous mills and the effect of grate design.{{Morrell and Stephenson (1996)}}
:<math>L_{\rm V} = m_1 \left ( \dfrac{F}{V} \right )^{m_2}</math>


2012, Kojovic, Updated Julius Kruttschnitt Semi - Autogenous Grinding Mill Model.{{Kojovic et al. (2012)}}
where:
* <math>L_{\rm V}</math> is the fraction of mill volume occupied by below grate size solids and water (v/v)
* <math>F</math> is the volumetric flow rate of slurry discharged from the mill (m<sup>3</sup>/min)
* <math>m_1</math> is a constant related to grate design and mill speed
* <math>m_2</math> is a constant assumed to take the value of 0.5.{{Kojovic et al. (2012)}}


[[Tumbling Mill (Slurry Flow)]]
The volume of the mill, <math>V</math>, is calculated as the sum of a cylinder and two right circular frustums:{{Gupta and Yan (2016)}}
 
:<math>V = \pi {R_{\rm m}}^2L + 2 \cdot \bigg[ \dfrac{\pi}{3} (R_{\rm m} - R_{\rm t}) \cdot \tan \alpha_{\rm c} \cdot \left ( {R_{\rm m}}^{2} + R_{\rm m} R_{\rm t} + {R_{\rm t}}^{2} \right) \bigg]</math>
 
where:
* <math>R_{\rm m}</math> is the radius of the mill inside the liners (m), equal to half of the diameter of the mill inside the liners, <math>D</math> (m)
* <math>R_{\rm t}</math> is the radius of the discharge trunnion (m), equal to half of the diameter of the discharge trunnion, <math>D_{\rm t}</math> (m)
* <math>L</math> is the length of the cylindrical (belly) section of the mill (m)
* <math>\alpha_{\rm c}</math> is the cone angle, measured as the angular displacement of the cone surface from the vertical direction (rad)
 
Morrell and Stephenson (1996) related discharge flow rate to slurry hold-up, grate design and mill speed with the following semi-empirical relation:{{Morrell and Stephenson (1996)}}
 
:<math>Q_{\rm m} = k_{\rm m} {J_{\rm pm}}^2 \gamma^{2.5} A \phi^{-1.34} D^{0.5}</math>
 
where
* <math>Q_{\rm m}</math> is the volumetric discharge rate of slurry through the grinding media zone  (m<sup>3</sup>/h)
* <math>k_{\rm m}</math> is the slurry discharge coefficient for the grinding media zone
* <math>J_{\rm pm}</math> is the net fractional slurry hold-up in the grinding media interstices (v/v)
* <math>A</math> is the total open area of grate apertures (m<sup>2</sup>)
* <math>\phi</math> is the fraction critical speed of the mill (frac)
* <math>D</math> is the mill diameter (m)
 
and the mean radial position of the grate apertures, <math>\gamma</math> (m/m), is defined as:
 
:<math>\gamma = \frac{\sum{r_ia_i}}{r_{\rm m} \sum{a_i}}</math>
 
where:
* <math>a_i</math> is the open area of all holes (m<sup>2</sup>) at radial position <math>r_i</math> (m)
* <math>r_{\rm m}</math> is the radius of the mill inside the liners (m)
 
In addition, solids greater than diameter <math>x_{\rm m}</math> but less than the grate aperture size will also discharge from the mill. To estimate total discharge flow rate, <math>Q</math> (m<sup>3</sup>/h), Morrell and Stephenson (1996) suggest the following correction:
 
:<math>Q = k_{\rm g} Q_{\rm m}</math>
 
where <math>k_{\rm g}</math> is a factor to account for coarse material, and taking the values shown in Table 1.
 
:{| class="wikitable"
|+ Table 1. Applied values for <math>k_{\rm g}</math>.
|-
! Aperture !! <math>k_{\rm g}</math>
|-
| Grates only; <19mm || 1.07
|-
| Grates only; 19mm - 28mm|| 1.125
|-
| Grates >38mm or pebble ports || 1.2
|}
 
The value of <math>m_1</math> for a given grate design and mill can be determined by observing that <math>L_{\rm V} = J_{\rm pm}</math> and <math>F = Q \big / 60</math> in the above equations.{{Kojovic et al. (2012)}} That is:
 
:<math>m_1 \left ( \dfrac{Q}{60V} \right )^{0.5} = {k_{\rm g}}^{-0.5} {k_{\rm m}}^{-0.5}  Q^{0.5} \gamma^{-1.25} A^{-0.5} \phi^{0.67} D^{-0.25}</math>
 
The mill volume, <math>V</math>, can expressed as the product of the mill cross sectional area and an ''equivalent grinding length'', <math>L_{\rm eq}</math> (m), i.e.:
 
:<math>V = \pi \left ( \dfrac{D}{2} \right )^2 L_{\rm eq} \implies L_{\rm eq} = \dfrac{V}{\pi \left ( \frac{D}{2} \right )^2}</math>
 
Furthermore, the total open area of the grates, <math>A</math> (m<sup>2</sup>), can be replaced with an expression combining the grate ''open area fraction'',  <math>A_{\rm OF}</math> (m<sup>2</sup>/m<sup>2</sup>), and mill cross-sectional area:
 
:<math>A = \pi \left ( \dfrac{D}{2} \right )^2 A_{\rm OF}</math>
 
Replacing the <math>V</math> and <math>A</math> terms in the <math>m_1</math> equation above yields:
 
:<math>m_1 = \sqrt{60} \cdot ({k_{\rm g}} {k_{\rm m}})^{-0.5} \gamma^{-1.25} {A_{\rm OF}}^{-0.5} \phi^{0.67} D^{-0.25} {L_{\rm eq}}^{0.5}</math>
 
Thus, slurry hold-up can be computed for a given feed/discharge flow rate, grate design and mill.
 
=== Discharge rates ===
 
[[File:AGSAGVariableRates7.png|thumb|425px|Figure X. Classification function, <math>C_i</math>, with pebble port open are fraction, <math>f_p</math>, specified.]]
 
[[File:AGSAGVariableRates7.png|thumb|425px|Figure X. Classification function, <math>C_i</math>, where pebble port open are fraction, <math>f_p</math>, is zero, i.e grates only.]]
 
The discharge rate of solids from the grate of a perfectly mixed mill is:{{Kojovic et al. (2012)}}
 
:<math>p_i = D_i.s_i</math>
 
where:
:<math>D_i = d_{\rm max}.C_i</math>
:<math> C_i =
    \begin{cases}
      1 & \bar d_i \leq x_{\rm m}\\
      \left ( 1 - \dfrac{\ln \bar d_i -\ln x_{\rm m}}{\ln x_{\rm g} - \ln x_{\rm m} } \right )(1 - f_{\rm p}) + f_{\rm p} & x_{\rm m}<\bar d_i\leq x_{\rm g}\\   
      \left ( 1 - \dfrac{\ln \bar d_i -\ln x_{\rm g}}{\ln x_{\rm p} - \ln x_{\rm g} } \right ) .f_{\rm p}  & x_{\rm g}<\bar d_i\leq x_{\rm p}\\   
      0 & \bar d_i>x_{\rm p}\\
    \end{cases}
</math>
 
and:
* <math>d_{\rm max}</math> is the fraction of load presented to the mill discharge per unit of time (h<sup>-1</sup>)
* <math>C_{i}</math> is the classification function, the fraction of particles of size <math>i</math> reporting to the mill product (frac)
* <math>\bar d_i</math> is the [[Conversions|geometric mean size]] of particles in size interval <math>i</math> (mm)
* <math>x_{\rm m}</math> is the particle size below which all mass in the size interval reports to mill product (mm), i.e. like water
* <math>x_{\rm g}</math> is the grate aperture size (mm)
* <math>x_{\rm p}</math> is the pebble port size (mm)
* <math>f_{\rm p}</math> is the fraction of open area occupied by pebble ports (m<sup>2</sup>/m<sup>2</sup>)
 
Figure X shows an example classification function with pebble ports included, whilst Figure x shows the same function with a grate-only mill.
 
The value of <math>d_{\rm max}</math> is adjusted during the calculation sequence to ensure the fraction of solids less than <math>x_{\rm m}</math> plus water retained in the mill load computed by the [[AG/SAG Mill (Variable Rates)#Perfect_mixing_model|Perfect Mixing population balance]] matches the slurry hold-up determined by the [[AG/SAG Mill (Variable Rates)#Slurry_flow|slurry flow]] calculations.


=== Breakage rates ===
=== Breakage rates ===


1996, Morrell, Morrison, AG and SAG Mill Circuit Selection and Design by Simulation.{{Morrell and Morrison (1996)}}
[[File:AGSAGVariableRates7.png|thumb|425px|Figure X. Breakage rate distribution characterised by cubic spline interpolation.]]
 
The breakage rate at each size interval, <math>R_i</math> (h<sup>-1</sup>), is generated by [[Interpolation|cubic spline interpolation]] between five breakage rate knots (<math>R1 - R5</math>) at the 0.25, 4, 16, 44 and 128mm particle size positions.{{Leung et al. (1987)}}
 
Morrell and Morrison (1996) described the following set of empirical equations which relate the breakage rate knots <math>R1 - R5</math> to mill operating conditions:{{Morrell and Morrison (1996)}}
 
:<math>
\begin{array}{l}
\ln(R1) = \mathit{RConst}_1 + \dfrac{k_{11} + k_{12} \ln (R2) - k_{13} \ln (R3) + J_{\rm B} (k_{14} - k_{15} F_{80}) - D_{\rm B}}{S_{\rm b}}\\
\ln(R2) = \mathit{RConst}_2 + k_{21}+ k_{22}\ln(R3) - k_{23} \ln(R4) - k_{24} F_{80}\\
\ln(R3) = \mathit{RConst}_3 + S_{\rm a} + \dfrac{k_{31} + k_{32} \ln(R4) - k_{33} R_{\rm r}}{S_{\rm b}}\\
\ln(R4) = \mathit{RConst}_4 + S_{\rm b} (k_{41} + k_{42} \ln(R5) + J_{\rm B} (k_{43} - k_{44} F_{80}))\\
\ln(R5) = \mathit{RConst}_5 + S_{\rm a} + S_{\rm b} \left (k_{51} + k_{52} F_{80} + J_{\rm B} (k_{53} - k_{54} F_{80}) - 3.0 D_{\rm B} \right )\\
\end{array}
</math>
 
where:
* <math>\mathit{RConst}_1 - \mathit{RConst}_5</math> are user-defined constants which can be used to adjust modelled breakage rates to observed values
* <math>J_{\rm B}</math> is the fraction of total mill volume occupied by balls and associated voids (% v/v)
* <math>F_{80}</math> is the 80% passing size of new feed (mm)
* <math>k_{ij}</math> are the regression coefficients specified in Table 1


=== Discharge rates ===
:{| class="wikitable"
|+ Table 1. Breakage rate regression coefficients.
|-
! <math>j</math> !! <math>k_{1j}</math> !! <math>k_{2j}</math> !! <math>k_{3j}</math> !! <math>k_{4j}</math> !! <math>k_{5j}</math>
|-
| 1 || 2.504|| 4.682|| 3.141|| 1.057|| 1.894
|-
| 2 || 0.397|| 0.468|| 0.402|| 0.333|| 0.014
|-
| 3 || 0.597|| 0.327|| 4.632 || 0.171|| 0.473
|-
| 4 || 0.192|| 0.0085|| -|| 0.0014|| 0.002
|-
| 5 || 0.002 || - || -|| -|| -
|}
 
The mill rotational speed scaling factor, <math>S_{\rm a}</math>, is computed from the mill rotational speed (rpm) as:
 
:<math>S_{\rm a} = \ln \left ( \dfrac{\text{Mill speed (rpm)}}{23.6} \right )</math>
 
Similarly, the mill fraction critical speed scaling factor, <math>S_{\rm b}</math>, is computed from the [[Tumbling Mill (Speed)|mill fraction critical speed]], <math>\phi</math> (frac), as:
 
:<math>S_{\rm b} = \dfrac{\phi}{0.75}</math>
 
The ball diameter scaling factor, <math>D_{\rm B}</math>, is computed from the ball top size, <math>d_{\rm B}</math> (mm), as:
 
:<math>D_{\rm B} = \ln \left ( \dfrac{d_{\rm B}}{90} \right )</math>
 
The recycle ratio, <math>R_{\rm r}</math>, is the ratio of the mass flowrate of recycled -20+4mm material to the total mass flowrate of all new feed plus recycled -20+4mm material, i.e.
 
:<math>R_{\rm r} = \dfrac{Q_{\rm M,R} (P_{20\text{mm,R}} - P_{4\text{mm,R}})}{Q_{\rm M,F} + Q_{\rm M,R}(P_{20\text{mm,R}} - P_{4\text{mm,R}})}</math>


1987, Leung, An energy based ore specific model for autogenous and semi-autogenous grinding, Copper '87.{{Leung et al. (1987)}}
where:
* <math>Q_{\rm M,F}</math> is new feed mass flow rate (t/h)
* <math>Q_{\rm M,R}</math> is recycle feed mass flow rate (t/h)
* <math>P_{20\text{mm,R}}</math> and <math>P_{4\text{mm,R}}</math> are the fraction of recycle feed passing 20mm and 4mm size, respectively (frac)


2012, Kojovic, Updated Julius Kruttschnitt Semi - Autogenous Grinding Mill Model.{{Kojovic et al. (2012)}}
The recycle ratio, <math>R_{\rm r}</math>, is only applicable when the coarse recycled feed component consists of mill pebbles (scats) which have not undergone an intervening breakage step such as pebble crushing.


=== Specific comminution energy ===
=== Specific comminution energy ===
S20 units in mm.{{Leung et al. (1987)}}
:<math>S_{20} = \left (P_{100} \cdot P_{98} \cdot P_{96} \dots P_{80} \right )^{\frac{1}{11}} </math>
Need to show selection of top size mesh for P100 versus other Ps.
Energy level is potential energy of particle at S20 characteristic size and density falling through a the full height of the mill.{{{Leung et al. (1987)}} Energy level corrected from Leung for S20 radius in volume calc and units to kWh:
:<math>\mathit{EL} = \dfrac{\frac{4}{3} \pi \left ( \frac{S_{20}}{2000} \right )^3 \rho g D}{3600}</math>
Ecs.{{Bueno et al. (2013)}}
:<math>
(E_{\rm cs})_i =
\begin{cases}
\dfrac{\mathit{EL}}{\frac{4}{3} \pi \left ( \frac{\bar d_i}{2000} \right )^3 \rho} & i = 1\\
\dfrac{(E_{\rm cs})_1}{ \left ( \frac{\bar d_i}{\bar d_1} \right )^{1.5}  } & 1 < i \leq n
\end{cases}
</math>


1987, Leung, An energy based ore specific model for autogenous and semi-autogenous grinding, Copper '87.{{Leung et al. (1987)}}
1987, Leung, An energy based ore specific model for autogenous and semi-autogenous grinding, Copper '87.{{Leung et al. (1987)}}


=== Equivalent ball size ===
S20 calc here.
 
Then E1 (or EL?) to Ecs per size.
 
=== Appearance function ===
 
==== High energy ====
 
Also reference crusher model tn splines etc. But always fixed spline for this model.
 
==== Low energy ====
 
==== Combined energy ====


TODO


=== Ball load ===
=== Ball load ===


TODO
Ball sizing root 2 convereted to full size, interpolation etc.


Equivalent ball size
Converts ball load to equivalent ore for S20 calc.
If no ball load, autogenous milling.
=== Internal mesh series ===
=== Multicomponent modelling ===
=== Mill power ===
Show equations for Jt etc to get inputs for Morrell power model.
Default uses Morrell 1992, for info only. Other models can be applied, not directly used in calculations.
=== Additional notes ===
An important, and potentially overlooked, limitation of the Variable Rates model is the insensitivity of the breakage rate relationships to mill load. Only valid at constant load, note about this.
As-published model does not include slurry pool. Model excludes slurry pool in slurry flow and power model.
-->
-->


Line 60: Line 311:
{{Excel (Text, Inputs)}}
{{Excel (Text, Inputs)}}


:<math>Parameters=
:<math>\mathit{Parameters} =
\begin{bmatrix}
\begin{bmatrix}
D\text{ (m)}\\
D\text{ (m)}\\
Line 67: Line 318:
\alpha_{c}\text{ (deg.)}\\
\alpha_{c}\text{ (deg.)}\\
\phi\text{ (frac)}\\
\phi\text{ (frac)}\\
A\text{ (m}^{\text{2}}\text{)}\\
A_{\rm OF}\text{ (m}^{\text{2}}\text{)}\\
f_{\rm p}\text{ (m}^2\text{/m}^2\text{)}\\
f_{\rm p}\text{ (m}^2\text{/m}^2\text{)}\\
x_{\rm p}\text{ (mm)}\\
x_{\rm p}\text{ (mm)}\\
Line 86: Line 337:
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


Size = \begin{bmatrix}
\mathit{Size} = \begin{bmatrix}
d_{1}\text{ (mm)}\\  
d_{1}\text{ (mm)}\\  
\vdots\\  
\vdots\\  
Line 92: Line 343:
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


MillNewFeed= \begin{bmatrix}
\mathit{MillNewFeed} = \begin{bmatrix}
(Q_{\rm M,F})_{11}\text{ (t/h)} & \dots & (Q_{\rm 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\\  
Line 98: Line 349:
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


MillRecycleFeed= \begin{bmatrix}
\mathit{OreSG} = \begin{bmatrix}
(Q_{\rm M,R})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,R})_{1m}\text{ (t/h)}\\
(\rho_{\rm S})_{1}\text{ (t/m}^\text{3}\text{)} & \dots & (\rho_{\rm S})_m\text{ (t/m}^\text{3}\text{)}\\  
\vdots & \ddots & \vdots\\
\end{bmatrix},\;\;\;\;\;\;
(Q_{\rm M,R})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,R})_{nm}\text{ (t/h)}\\  
</math>
\end{bmatrix}^*


</math>


:<math>
:<math>
OreSG= \begin{bmatrix}
\mathit{BallSizing} = \begin{bmatrix}
(\rho_{\rm S})_{1}\text{ (t/m}^\text{3}\text{)} & \dots & (\rho_{\rm S})_m\text{ (t/m}^\text{3}\text{)}\\
(\mathit{MF}_{\rm B})_{d_{\rm B}/\sqrt{2}}\text{ (}%\text{ w/w)}\\
\end{bmatrix},\;\;\;\;\;\;
(\mathit{MF}_{\rm B})_{d_{\rm B}/(2\sqrt{2}})\text{ (}%\text{ w/w)}\\
 
(\mathit{MF}_{\rm B})_{d_{\rm B}/(3\sqrt{2}})\text{ (}%\text{ w/w)}\\
BallSizing = \begin{bmatrix}
(\mathit{MF}_{\rm B})_{d_{\rm B}/(4\sqrt{2}})\text{ (}%\text{ w/w)}\\
({\rm MF}_{\rm B})_{d_{\rm B}/\sqrt{2}}\text{ (}%\text{ w/w)}\\
({\rm MF}_{\rm B})_{d_{\rm B}/(2\sqrt{2}})\text{ (}%\text{ w/w)}\\
({\rm MF}_{\rm B})_{d_{\rm B}/(3\sqrt{2}})\text{ (}%\text{ w/w)}\\
({\rm MF}_{\rm B})_{d_{\rm B}/(4\sqrt{2}})\text{ (}%\text{ w/w)}\\
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


RConst= \begin{bmatrix}
\mathit{RConst} = \begin{bmatrix}
{\rm RConst}_1\\  
\mathit{RConst}_1\\  
\vdots\\  
\vdots\\  
{\rm RConst}_k\\  
\mathit{RConst}_5\\  
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


OreBreakageParams=
\mathit{OreBreakageParams} =
\begin{bmatrix}
\begin{bmatrix}
A_1 & \dots & A_k\\
A_1 & \dots & A_m\\
b_1 & \dots & b_k\\
b_1 & \dots & b_m\\
(t_{\rm a})_1 & \dots & (t_{\rm a})_k\\
(t_{\rm a})_1 & \dots & (t_{\rm a})_m\\
\end{bmatrix}  
\end{bmatrix},\quad
 
\mathit{MillRecycleFeed} = \begin{bmatrix}
(Q_{\rm M,R})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,R})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,R})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,R})_{nm}\text{ (t/h)}\\
\end{bmatrix}^*
</math>
</math>


where:
where:
* <math>D</math> is the diameter of the mill inside the liners (m)
* <math>L</math> is the length of the belly section of the mill (m)
* <math>D_{\rm t}</math> is diameter of the discharge trunnion (m)
* <math>\alpha_{\rm c}</math> is angle between the cone end surface and the vertical direction (deg)
* <math>\phi</math> is the fraction critical speed of the mill (frac)
* <math>A</math> is the open area fraction of the discharge grate (frac)
* <math>f_{\rm p}</math> is the pebble port area fraction (frac)
* <math>f_{\rm p}</math> is the pebble port area fraction (frac)
* <math>x_{\rm p}</math> is the size of the pebble port aperture (mm)
* <math>x_{\rm p}</math> is the size of the pebble port aperture (mm)
* <math>x_{\rm g}</math> is the size of the grate aperture (mm)
* <math>x_{\rm g}</math> is the size of the grate aperture (mm)
* <math>x_{\rm m}</math> is the fine size (mm)
* <math>x_{\rm m}</math> is the fine size (mm)
* <math>\gamma</math> is the mean radial position of the grate apertures (m/m)
* <math>k</math> is the slurry discharge coefficient
* <math>J_{\rm B}</math> is the ball load volume fraction (v/v)
* <math>J_{\rm B}</math> is the ball load volume fraction (v/v)
* <math>\rho_{\rm B}</math> is the density or specific gravity of the ball media (t/m<sup>3</sup> or -)
* <math>\rho_{\rm B}</math> is the density or specific gravity of the ball media (t/m<sup>3</sup> or -)
* <math>d_{\rm B}</math> is ball top size (mm)
* <math>(F_{80})_{\rm Ref}</math> is the reference ''F<sub>80</sub>'' size (mm)
* <math>(F_{80})_{\rm Ref}</math> is the reference ''F<sub>80</sub>'' size (mm)
* <math>(d_1)_{\rm Int}</math> is the internal mesh top size (mm)
* <math>(d_1)_{\rm Int}</math> is the internal mesh top size (mm)
Line 159: Line 401:
* <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_{\rm M,F}</math> is new feed mass flow rate (t/h)
* <math>Q_{\rm M,R}</math> is recycle feed mass flow rate (t/h)
* <math>\rho_{\rm S}</math> is the density or specific gravity of solids (t/m<sup>3</sup> or -)
* <math>\rho_{\rm S}</math> is the density or specific gravity of solids (t/m<sup>3</sup> or -)
* <math>{\rm MF}_{\rm B}</math> is the mass fraction retained of balls (% w/w)
* <math>\mathit{MF}_{\rm B}</math> is the mass fraction retained of balls (% w/w)
* <math>k</math> is the number of breakage rate knots
* <math>{\rm RConst}</math> is a breakage rate constant
* <math>A</math> (%), <math>b</math>, and <math>t_{\rm a}</math> are ore breakage parameters
* <math>A</math> (%), <math>b</math>, and <math>t_{\rm a}</math> are ore breakage parameters
* <math>^*</math> indicates the <math>\mathit{MillRecycleFeed}</math> array is an optional input parameter, and is set to null if omitted
* <math>^*</math> indicates the <math>\mathit{MillRecycleFeed}</math> array is an optional input parameter, and is set to null if omitted
Line 193: Line 431:
d_{\rm Max}\text{ (h}^{-1}\text{)}\\
d_{\rm Max}\text{ (h}^{-1}\text{)}\\
S_{20}\text{ (mm)}\\
S_{20}\text{ (mm)}\\
{\rm EL}\text{ (kWh)}\\
\mathit{EL}\text{ (kWh)}\\
\rho_{\rm c}\text{ (t/m}^3\text{)}\\
\rho_{\rm c}\text{ (t/m}^3\text{)}\\
\theta_{\rm S}\text{ (rad)}\\
\theta_{\rm S}\text{ (rad)}\\
Line 697: Line 935:


{{SysCAD (Page, About)}}
{{SysCAD (Page, About)}}
== See also ==
* [[Tumbling Mill (Slurry Flow)]]


== References ==
== References ==

Revision as of 03:20, 20 August 2024

Description

This article describes an implementation of the Autogenous (AG) and Semi-Autogenous (SAG) mill model originated by Leung (1987) and extended with variable breakage rates by Morrell and Morrison (1996).[1][2][3]

The formulation is referred to in the associated literature as the "Variable Rates" model (Morrell et al., 2001).[4]

Model theory

Under construction icon-blue.svg.png This section is currently under construction. Please check back later for updates and revisions.

Excel

The Variable Rates AG/SAG mill model may be invoked from the Excel formula bar with the following function call: 

=mdUnit_AGSAG_VariableRates(Parameters as Range, Size as Range, MillNewFeed as Range, OreSG as Range, BallSizing as Range, RConst as Range, OreBreakageParams as Range, Optional MillRecycleFeed as Range = Nothing)

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 the pebble port area fraction (frac)
  • is the size of the pebble port aperture (mm)
  • is the size of the grate aperture (mm)
  • is the fine size (mm)
  • is the ball load volume fraction (v/v)
  • is the density or specific gravity of the ball media (t/m3 or -)
  • is the reference F80 size (mm)
  • is the internal mesh top size (mm)
  • is the charge void fraction (v/v)
  • 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 an index of the Appearance function to view in the results
  • is an index of the Appearance function to view in the results
  • is the number of intervals
  • is the number of ore types
  • 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 density or specific gravity of solids (t/m3 or -)
  • is the mass fraction retained of balls (% w/w)
  • (%), , and are ore breakage parameters
  • indicates the array is an optional input parameter, and is set to null if omitted

Results

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

where:

  • is the number of internal computation steps required to converge the load
  • is the numerical error of the converged load approximation
  • 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 mass of ore solids in the mill (t)
  • is the mass of liquids in the mill (t)
  • is the mass of balls in the mill (t)
  • is the total mass of ore, liquids and balls in the mill (t)
  • is the charge volume fraction (v/v)
  • is the void fill fraction (v/v)
  • is the coarse factor (-)
  • is a parameter of the Austin mill holdup equation
  • is a parameter of the Austin mill holdup equation
  • is maximum discharge rate from the mill (h-1)
  • is the geometric mean size of the top 20% of the load (mm)
  • is the energy level in the mill (kWh)
  • is the charge density (t/m3)
  • is angular position of the charge shoulder (rad)
  • is angular position of the charge toe (rad)
  • is the charge surface radius (m)
  • is the no-load power of the mill (kW)
  • is the net power of the mill (kW)
  • is the gross power of the mill (kW)
  • is product mass flow rate (t/h)
  • is the geometric mean size of the internal mesh series interval that mass is retained on (mm)
  • is the discharge rate (h-1)
  • is the breakage rate (h-1)
  • is specific comminution energy (kWh/t)
  • is the Appearance function (frac)

Example

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

Figure 1. Example showing the selection of the Parameters (blue frame) array in Excel.
Figure 2. Example showing the selection of the Size (dark red frame), OreSG (green frame), MillNewFeed (purple frame) and MillRecycleFeed (light red frame) arrays in Excel.
Figure 3. Example showing the selection of the BallSizing (purple frame), RConst (brown frame), and OreBreakageParams (teal frame) arrays in Excel.
Figure 8. 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.

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.
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.
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
VariableRates
HelpLink ButtonModelHelp.png Opens a link to this page using the system default web browser. Note: Internet access is required.
Requirements
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.
Mill
Diameter Input The inside liner diameter of the mill.
BellyLength Input The inside liner belly length of the mill, excluding cones.
TrunnionDiameter Input The inside liner trunnion diameter of the mill.
ConeAngle Input Angle of the feed and discharge end cones, measured as positive displacement from the vertical direction.
FracCS Input The fraction critical speed of the mill.
Grate
OpenAreaFrac Input Open area fraction of the grate.
PebblePortFrac / fp Input Pebble port area fraction.
PebblePortAperture / xp Input Pebble port aperture size.
GrateAperture / xg Input Grate aperture size.
FineSize / xm Input Fine size, size at which particles behave like water.
MeanRadialPosition / gamma Input Mean radial position of the grate apertures.
SlurryDischCoeff / k Input Slurry discharge coefficient.
Ball
BallLoad Input Ball load fraction.
BallSG Input Density (Specific Gravity) of ball media.
BallTopSize Input Top size of new ball media.
Size Display Ball sizing intervals.
Load Display Mass fraction retained of ball media in each ball sizing interval.
RFunction
RSize Display Spline knot positions.
RConst Input Values of at each spline knot position.
Other
ReferenceF80 Input Reference F80 size.
InternalMeshTopSize Input Top size of internal mesh series.
Voidage Input Volumetric fraction of void space in charge.
NetPowerAdjust Input Net Power Adjust factor of mill power equation.

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
Ore
OreSpecific CheckBox
  • Ore-specific parameters, allows the ore breakage parameters 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.
A Input / Display Impact ore breakage parameter.
b Input / Display Impact ore breakage parameter.
ta Input / Display Abrasion ore breakage parameter.
AGSAGVariableRates5.png AGSAGVariableRates6.png

Results page

This page is used to display the model results.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Results
Solver
Iterations Display Number of iterations to converge internal load solver.
IterationError Display Numerical approximation error of internal load solver.
Mill Properties
MillVolume Display Internal volume of the mill.
MillSpeed Display Rotational speed of the mill.
MillFeedRate / Feed.SLQv Display Volumetric feed rate of pulp into the mill.
Mill Contents
OreMass Display Mass of ore (solids with PSD) in the mill.
LiquidMass Display Mass of liquids in the mill.
BallMass Display Mass of ball media in the mill.
TotalChargeMass Display Total mass of ore, liquids and balls in the mill.
VolTotalLoad Display Volumetric fraction of mill volume of total charge (ore, liquids, balls and void space).
Mill Discharge
m1 Display Parameter of the Austin mill holdup relationship.
m2 Display Parameter of the Austin mill holdup relationship.
dMax Display Maximum discharge rate of load volume through the grate.
Charge Properties
S20 Display Size of the top (largest) 20% of the load.
ChargeDensity Display Density of the charge.
U Display Fraction of charge void space filled with slurry.
ThetaShoulder Display Angular position of the charge shoulder.
ThetaTue Display Angular position of the charge toe.
ChargeSurfaceRadius Display Radius of the inner charge surface.
Power
NoLoadPower Display No-load power draw of the mill.
NetPower Display Net power draw of the mill.
GrossPower Display Gross power draw of the mill.

RiDi page

This page displays the breakage and discharge rates for each size interval computed by the model.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Rates
Size Display Size of each interval in internal mesh series.
MeanSize Display Geometric mean size of each interval in internal mesh series.
R Display Value of breakage rate, , for each size interval, for each ore species.
D Display Value of discharge rate, , for each size interval.
Ecs Display Value of the specific comminution energy for each size interval.

Load page

This page displays information about the balls, solids and liquids that currently comprise the mill load.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Distribution
Name Display Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream.
IntervalCount Display Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream.
SpWithPSDCount Display Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
Load
SolidMass / SMt Display The mass of solids with the SysCAD size distribution property currently in the mill.
LiquidMass / LMt Display The mass of liquids plus solids without the SysCAD size distribution property currently in the mill.
BallMass / BMt Display The mass of ball media in the mill.
Size Display Size of each interval in the external mesh series.
MeanSize Display Geometric mean size of each interval in the external mesh series.
Load Display The mass of solids with the SysCAD size distribution property currently in the mill, by size and 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.

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.
  2. ↑ Leung, K., Morrison, R.D. and Whiten, W.J., 1987. An Energy Based Ore Specific Model for Autogenous and Semi-autogenous Grinding, Copper 87, Vina del Mar, Vol. 2, pp 71 - 86
  3. ↑ Morrell, S. and Morrison, R.D., 1996. AG and SAG mill circuit selection and design by simulation. In International Conference on Autogenous and Semiautogenous Grinding Technology (Vol. 2, pp. 769-790).
  4. ↑ Morrell, S., Valery, W., Banini, G. and Latchireddi, S., 2001. Developments in AG/SAG mill modelling. Proceedings of Autogenous and Semiautogenous Grinding Technology, Vancouver, pp.71-84.
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