View source for AG/SAG Mill (Variable Rates)

== 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).{{Napier-Munn et al. (1996)}}{{Leung et al. (1987)}}{{Morrell and Morrison (1996)}}

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

== Model theory ==

[[File:AGSAGVariableRates7.png|thumb|450px|Figure 1. Schematic diagram of internal AG/SAG mill processes (after Napier-Munn et al., 1996).{{Napier-Munn et al. (1996)}}]]

[[File:AGSAGVariableRates8.png|thumb|450px|Figure 3. Variable Rates AG/SAG mill model iterative calculation sequence (after Napier-Munn et al., 1996).{{Napier-Munn et al. (1996)}}]]

The schematic diagram in Figure 1 illustrates the primary processes of ''feed'', ''breakage'', ''classification'' and ''discharge'' occurring within AG/SAG mills.

The Variable Rates AG/SAG model ties these processes together through the ''perfect mixing model'', which 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 2 below represents the balance for a given size fraction:

::::{|
| style="padding: 10px"|<gallery mode="nolines" widths=950px heights=36px>
File:BallMillPerfectMixing1.png|Figure 2. Schematic diagram of the steady-state population balance adopted by the Variable Rates AG/SAG 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 volumetric flow rate of solids of size interval <math>i</math> in the mill feed (m<sup>3</sup>/h)
* <math>p_i</math> is the volumetric flow rate of solids of size interval <math>i</math> in the mill product (m<sup>3</sup>/h)
* <math>s_i</math> is the volume of solids of size interval <math>i</math> in the mill load (m<sup>3</sup>)
* <math>R_i</math> is the breakage rate of solids of size interval <math>i</math> in the mill load  (h<sup>-1</sup>)
* <math>A_{ij}</math> is the appearance function, the distribution of particle volume arising from the breakage of a parent particle in size interval <math>j</math> into progeny of size interval <math>i</math> (frac)

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 (h<sup>-1</sup>).

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 L} - D_{\rm L} s_{\rm L} = 0</math>

where:
* <math>s_{\rm L}</math> is the load volume of liquids in the mill (m<sup>3</sup>)
* <math>f_{\rm L}</math> is the volumetric feed rate of liquids into the mill (m<sup>3</sup>/h)
* <math>D_{\rm L}</math> is the discharge rate of liquids from the mill, normally assumed to equal <math>D_n</math> (h<sup>-1</sup>).

=== Calculation sequence ===

The Variable Rates AG/SAG model uses a range of sub-models to quantify the breakage rate (<math>R</math>), appearance function (<math>A</math>), and discharge function (<math>D</math>) terms of the perfect mixing model, and hence compute the mill load (<math>s</math>) and product (<math>p</math>) at steady-state.

These sub-models are strongly interactive, and an iterative calculation sequence is necessary to numerically solve the steady-state population balance. The required calculation sequence is presented in Figure 3.

The sub-models are described in further detail below.

=== Breakage rates ===

[[File:AGSAGVariableRates12.png|thumb|450px|Figure 4. 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 128 mm 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.

:{| class="wikitable"
|+ Table 1. Breakage rate regression coefficients (after Morrell and Morrison, 1996).{{Morrell and Morrison (1996)}}

|-
! <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,Top}</math> (mm), as:

:<math>D_{\rm B} = \ln \left ( \dfrac{d_{\rm B,Top}}{90} \right )</math>

The recycle ratio, <math>R_{\rm r}</math>, is the ratio of the mass flowrate of recycled -20+4 mm material to the total mass flowrate of all new feed plus recycled -20+4 mm 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>

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 20 mm and 4 mm size, respectively (frac)

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.

Figure 4 presents an example breakage rate distribution constructed from the five breakage rate knots and a continuous cubic spline.

=== Discharge rates ===

The discharge rates (<math>D</math>) are related to the hold-up of slurry in the mill and particle classification at the discharge grates.

==== Slurry hold-up ====

[[File:AGSAGVariableRates9.png|thumb|450px|Figure 5. Principal dimensions of an AG/SAG mill.]]

The volumetric flow rate of slurry discharged from a grated mill depends on the level of 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)}}

:<math>L_{\rm V} = m_1 \left ( \dfrac{F}{V} \right )^{m_2}</math>

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)}}

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)

The principal dimensions of an AG/SAG mill required to compute mill volume and other properties are shown in Figure 5.

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 to fine slurry, particles up to the 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:{{Morrell and Stephenson (1996)}}

:<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 2.

:{| class="wikitable"
|+ Table 2. Applied values for <math>k_{\rm g}</math>.{{Morrell and Stephenson (1996)}}
|-
! 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, i.e.:{{Kojovic et al. (2012)}}

:<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, <math>L_{\rm V}</math>, can be computed for a given feed/discharge flow rate, grate design and mill.

==== Classification and discharge ====

[[File:AGSAGVariableRates10.png|thumb|450px|Figure 6. Classification function, <math>C_i</math>, with pebble port open are fraction, <math>f_p</math>, specified.]]

[[File:AGSAGVariableRates11.png|thumb|450px|Figure 7. 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 6 shows an example classification function with pebble ports included, whilst Figure 7 shows the same function with a grate-only mill.

The value of <math>d_{\rm max}</math> is adjusted during the calculation sequence (Figure 3) to ensure the fraction of solids less than <math>x_{\rm g}</math> plus water retained in the mill load computed by the perfect mixing population balance matches the slurry hold-up determined by the [[AG/SAG Mill (Variable Rates)#Slurry_flow|slurry flow]] calculations.

=== Appearance function ===

The appearance function, <math>A</math>, is defined as the mass-by-size distribution of progeny particles resulting from the breakage of parent particles.

Two types of particle breakage are theorised to occur occur within AG/SAG mills:

# ''High energy'' breakage from the impact of cataracting balls and large rocks at the toe of the charge, and
# ''Low energy'' breakage from the abrasion and attrition of particles within the charge bed.

The appearance function for an AG/SAG mill is constructed from the combination of appearance functions for the high energy, <math>A_{\rm HE}</math>, and low energy, <math>A_{\rm LE}</math>, breakage regimes.

An appearance function is a lower triangular matrix as all broken particles are, by definition, smaller than their parent particle.

==== High energy ====

Leung et al. (1987) related the amount of energy available for high energy impact breakage in a mill to the mean size of the top 20% of the charge, <math>S_{20}</math> (mm). The <math>S_{20}</math> is defined as:{{Leung et al. (1987)}}

:<math>S_{20} = \left (P_{100} \cdot P_{98} \cdot P_{96} \dots P_{80} \right )^{\frac{1}{11}} </math>

where <math>P_{x}</math> is the size passing fraction <math>x</math> (mm) of the charge, which includes ore and balls (if present).

A particle of representative size <math>S_{20}</math> falling through the full height of the mill converts the following potential energy into impact (kinetic) energy:

:<math>E_q = \dfrac{\frac{4}{3} \pi \left ( \dfrac{S_{20}}{2000} \right )^3 \rho_{\rm S} g D}{3600}</math> 

where:
* <math>E_q</math> is the energy level in the mill applied to particles in size fraction <math>q</math> (kWh)
* <math>q</math> is defined as the index of the size interval containing the largest particle in the feed, <math>1 \leq q \leq n</math>
* <math>\rho_{\rm S}</math> is the density of solids in the mill (t/m<sup>3</sup>)
* <math>g</math> is acceleration due to gravity (m/s<sup>2</sup>)
* <math>D</math> is the mil diameter (m)

and the above equation has been corrected from Leung et al. (1987) for particle radius and measurement units.

The energy level, <math>E_q</math>, is converted to the specific comminution energy applied to a particle in size fraction <math>q</math>, <math>(E_{\rm cs})_q</math> (kWh/t), by:{{Bueno et al. (2013)}}

:<math>(E_{\rm cs})_q = \dfrac{E_p}{\frac{4}{3} \pi \left ( \dfrac{\bar d_q}{2000} \right )^3 \rho_{\rm S}}</math> 

where <math>\bar d_q</math> is the [[Conversions#Geometric_mean_size|geometric mean size]] (mm) of a particle in size interval <math>q</math>, and the equation is again corrected for radius and units.

The specific comminution energy <math>(E_{\rm cs})_q</math> is scaled to smaller size fractions by:{{Bueno et al. (2013)}}

:<math>{(E_{\rm cs})_i} = 
\begin{cases}
0 & i < q\\
\dfrac{(E_{\rm cs})_q}{ \left ( \dfrac{\bar d_i}{\bar d_q} \right )^{1.5}  } & q \leq i \leq n\\
\end{cases}
</math>

To compute the high energy appearance function, the fraction of particles passing one-tenth of the initial mean particle, the <math>t_{\rm 10}</math> (frac), is determined for the specific comminution energy applied at each size:{{Leung et al. (1987)}}

:<math>(t_{10})_i = A(1-{\rm e}^{-b (E_{\rm cs})_i})</math>

where <math>A</math> and <math>b</math> are ore-specific hardness parameters obtained from a drop weight test.{{Napier-Munn et al. (1996)}}

The Variable Rates AG/SAG model uses the standard matrix of cumulative fraction passing data points in Table 3 to describe the products of a high breakage event for any ore type.{{Napier-Munn et al. (1996)}} 

Each element of the matrix, <math>t_{x,y}</math>, represents the percentage fraction of progeny particles passing one-<math>y</math>th of the original parent particle geometric mean size, when <math>x</math>% of the products pass one-tenth of the original product size (i.e. the <math>t_{10}</math>).

:{| class="wikitable"
|+ Table 3. Standard appearance function data used in the Variable Rates AG/SAG model.{{Napier-Munn et al. (1996)}}
!
!<math>t_{75}(\%)</math>
!<math>t_{50}(\%)</math>
!<math>t_{25}(\%)</math>
!<math>t_{4}(\%)</math>
!<math>t_{2}(\%)</math>
|-
|<math>t_{10}=10\%</math>
|2.33
|3.06
|4.98
|23.33
|50.53
|-
|<math>t_{10}=20\%</math>
|6.89
|9.41
|15.62
|61.58
|92.49
|-
|<math>t_{10}=50\%</math>
|10.32
|14.71
|25.88
|82.86
|96.47
|}

This matrix allows a complete breakage product size distribution for any parent size and mesh series to be reconstructed from the <math>t_{10}</math> via the following steps:

# A [[Interpolation#Cubic_spline_interpolation|cubic spline]] is used to interpolate <math>t_{75}</math> - <math>t_2</math> values for the <math>(t_{\rm 10})_i</math> at each particle size fraction's specific comminution energy. 
# A secondary cubic spline interpolation is used to produce a distribution across the full mesh size interval range for each parent particle size.

The [[Crusher (Whiten)|Whiten crusher]] model also applies a similar appearance function spline interpolation procedure.

==== Ball load ====

Semi-Autogenous (SAG) mills add steel ball grinding media to the ore charge to increase size reduction.

The mass of balls in a mill, <math>M_{\rm B}</math> (t), is related to the volume occupied by the balls by:{{Morrell (1992)}}

:<math>M_{\rm B} = J_{\rm B} (1 - \varepsilon) V \rho_{\rm B}</math>

where 
* <math>J_{\rm B}</math> is the fraction of mill volume occupied by balls and the void space between them, when the mill is at rest (v/v)
* <math>\varepsilon</math> is the void fraction, taken as 0.4 (v/v)

The calculation of specific comminution energy for the [[AG/SAG Mill (Variable Rates)#High_energy|high energy appearance function]] above uses the average size of the top 20% of the charge, the <math>S_{20}</math>. A charge that includes balls must incorporate the media load into the calculation of the <math>S_{20}</math>. This is achieved by considering the ball load as if each ball were an ore particle with the same mass, but a larger diameter to compensate for the density difference between steel and rock, i.e.:{{Leung et al. (1987)}}

:<math>
m_{\rm B} = v_{\rm B} \rho_{\rm B} = \frac{4}{3} \pi \left ( \frac{d_{\rm B}}{2} \right )^3 \rho_{\rm B}, \quad 
v_{\rm S} = \frac{4}{3} \pi \left ( \frac{d_{\rm S}}{2} \right )^3 = \frac{m_{\rm B}}{\rho_{\rm S}} = \frac{\frac{4}{3} \pi \left ( \frac{d_{\rm B}}{2} \right )^3 \rho_{\rm B}}{\rho_{\rm S}} \quad
\implies d_{\rm S} = \left ( \frac{{{d_{\rm B}}^3 \rho_{\rm B}}}{\rho_{\rm S}} \right )^{\frac{1}{3}}
</math>

where:
* <math>m_{\rm B}</math> and <math>v_{\rm B}</math> are the mass and volume of a ball, respectively.
* <math>\rho_{\rm B}</math> is the density of a ball 
* <math>d_{\rm B}</math> is the diameter of a ball 
* <math>v_{\rm S}</math> is the volume of an ore particle
* <math>\rho_{\rm S}</math> is the density of ore solids in the charge
* <math>d_{\rm S}</math> is the diameter of an ore particle with equivalent mass to a ball

Therefore, a weight fraction retained distribution of balls on any size interval series is incorporated into the the <math>S_{20}</math> calculation by the following steps:
# Scale the ball size intervals (<math>d_{\rm B}</math>) to equivalent ore size intervals (<math>d_{\rm S}</math>) as per the equation above.
# Use a [[Interpolation#Cubic_spline_interpolation|monotonic spline]] to [[Conversions#Convert_between_meshes|convert]] the ball distribution from the scaled equivalent ore size intervals (<math>d_{\rm S}</math>) to the size intervals used by the perfect mixing population balance formulation (<math>d_i</math>).
# Convert the weight fraction retained ball distribution to a mass retained distribution using the ball mass, <math>M_{\rm B}</math>.
# Convert the mass retained ball distribution to a volume retained distribution using the ore density, <math>\rho_{\rm S}</math>.
# Combine the ball volume-by-size and ore volume-by-size in the charge (<math>s_i</math>) when computing the <math>S_{20}</math>.

==== Low energy ====

The low energy appearance function is computed from the ore-specific abrasion parameter, <math>t_a</math> (%), which is obtained from the ore abrasion test described by Napier-Munn et al. (1996).{{Napier-Munn et al. (1996)}}

The cumulative fraction passing distribution of progeny particles arising from the abrasion breakage of a parent particle is computed by multiplying the <math>t_a</math> for a given ore by the scaling factors indicated in Table 4.{{Leung et al. (1987)}}

:{| class="wikitable"
|+ Table 4. Standard abrasion appearance function data.{{Leung et al. (1987)}}
|-
!  !! Cumulative fraction passing (%)
|-
| <math>t_{1}</math>|| <math>100</math>
|-
| <math>t_{1.25}</math>|| <math>2.687 \cdot t_{\rm a}</math>
|-
| <math>t_{1.5}</math>|| <math>1.631 \cdot t_{\rm a}</math>
|-
| <math>t_{10}</math>|| <math>1.0 \cdot t_{\rm a}</math>
|-
| <math>t_{100}</math>|| <math>0.9372 \cdot t_{\rm a}</math>
|-
| <math>t_{250}</math>|| <math>0.8070 \cdot t_{\rm a}</math>
|-
| <math>t_{500}</math>|| <math>0.6365 \cdot t_{\rm a}</math>
|}

The low energy appearance function is then determined for all parent particle sizes and across the full mesh series by spline interpolation of the cumulative fraction passing values computed above. 

Note that a [[Interpolation#Cubic_spline_interpolation|monotonic spline]] is employed for the low energy appearance function as the sharp change in cumulative fraction passing values between the original parent size (<math>t_1</math>) and the first progeny size (<math>t_{1.25}</math>) can result in undesired oscillation and negative function values when applying a regular cubic spline.

==== Combined appearance function ====

The high and low energy appearance functions are combined based on the relative proportions of <math>t_{\rm a}</math> and the <math>t_{10}</math> at each specific comminution energy:{{Leung et al. (1987)}}

:<math>A = \dfrac{t_{10} . A_{\rm HE} + t_{\rm a} . A_{\rm LE}}{t_{10} + t_{\rm a}}</math>

=== Mill power ===

The Variable Rates AG/SAG model includes an implementation of the [[Tumbling Mill (Power, Morrell Continuum)|Morrell Continuum]] tumbling mill power model. The predicted mill power draw is not utilised by the Variable Rates model formulation in any manner, and is provided for information only.

==== Charge properties ====

The power draw prediction requires an estimate of <math>J_{\rm t}</math> (v/v), the fraction of mill volume occupied by the charge, which includes coarse ore, balls, slurry, and void spaces.

Morrell (1992) provides relations for the mass of coarse ore, slurried ore, water and balls in a mill.{{Morrell (1992)}} Converting from mass to fraction of mill volume, these relations are:

:<math>\frac{V_{\rm co}}{V} = (J_{\rm t} - J_{\rm B})(1 - \varepsilon), \quad \frac{V_{\rm so}}{V} = J_{\rm t} \varepsilon U S, \quad \frac{V_{\rm L}}{V} = J_{\rm t} \varepsilon U (1 - S), \quad \frac{V_{\rm B}}{V} = J_{\rm B} (1 - \varepsilon)</math>

where:
* <math>V_{\rm co}</math>, <math>V_{\rm so}</math>, <math>V_{\rm L}</math>, and <math>V_{\rm B}</math> are the volumes of coarse ore, slurried ore, liquids, and balls in the mill, respectively (m<sup>3</sup>)
* <math>S</math> is the volume fraction of solids in the mill discharge (v/v)
* <math>U</math> is the fraction of void space between the coarse ore particles and balls that is filled with slurry (v/v)

The total volume of ore in the mill, <math>V_{\rm o}</math> (m<sup>3</sup>), is computed by the perfect mixing model, i.e.:

:<math>V_{\rm o} = \sum_{i=1}^n s_i</math>

Subtracting the slurried ore component from the total ore volume, the coarse ore volume is:

:<math>\frac{V_{\rm co}}{V} = \frac{V_{\rm o}}{V} - \frac{V_{\rm so}}{V} \implies (J_{\rm t} - J_{\rm B})(1 - \varepsilon) = \frac{\sum_{i=1}^n s_i}{V} - J_{\rm t} \varepsilon U S</math>

and rearranging for <math>J_{\rm t}</math> yields:

:<math>J_{\rm t} = \dfrac{J_{\rm B} (\varepsilon - 1) - \frac{\sum_{i=1}^n s_i}{V}}{\varepsilon (1 - U S) - 1}</math>

The value of <math>U</math> in the above equation is unknown, and is itself a function of <math>J_{\rm t}</math>. To simplify the calculations, <math>J_{\rm t}</math> is estimated for a given mill load and discharge by assuming <math>U = 1</math>.

Having computed an estimate for <math>J_{\rm t}</math>, the value of <math>U</math> may be approximated by:

:<math>U = \dfrac{L_{\rm V}}{\varepsilon J_{\rm t}}</math>

and the apparent density of the charge, <math>\rho_{\rm c}</math>, is:{{Morrell (1996a)}}

:<math>\rho_{\rm c} = \frac{J_{\rm t} \rho_{\rm S} (1 - \varepsilon + \varepsilon U S) + J_{\rm B}( \rho_{\rm B} - \rho_{\rm S})(1 - \varepsilon) + J_{\rm t} \varepsilon U (1 - S)}{J_{\rm t}}, \quad U \leq 1</math>

==== Power draw ====

:''Main article'': [[Tumbling Mill (Power, Morrell Continuum)]]

This implementation of the Variable Rates AG/SAG model applies Morrell's (1996) tumbling mill power draw equations for a grated mill which include terms for the conical end sections of the mill.{{Morrell (1996a)}}{{Bueno et al. (2013)}} 

The values of <math>J_{\rm t}</math>, <math>U</math>, and <math>\rho_{\rm c}</math>, plus mill dimensions and rotational speed are inputs to the power draw estimation equations. 

The equations return the following charge position and power draw results:
* The angular position of the charge shoulder, <math>\theta_{\rm S}</math> (rad)
* The angular position of the charge toe, <math>\theta_{\rm T}</math> (rad)
* The charge surface radius, <math>r_{\rm i}</math> (m)
* The no-load power of the mill, <math>P_{\rm NoLoad}</math> (kW)
* The net power of the mill, <math>P_{\rm Net}</math> (kW)
* The gross power of the mill, <math>P_{\rm Gross}</math> (kW)

The complete equations are excluded here for brevity and are available at the article link above.

=== Internal mesh series ===

The Variable Rates AG/SAG mill model is formulated internally with a geometric progression of <math>n = 42</math> mesh sizes at <math>\sqrt{2}</math> intervals, i.e.

:<math>d_i = 
\begin{cases}
d_1 & i=1\\
\dfrac{d_1}{i\sqrt{2}} & 1 < i < n\\
0 & i = n
\end{cases}
</math>

where <math>d_1</math> (mm) is the top size of the internal mesh series.

Feed, load and product size fractions are automatically converted to and from the internal mesh series during model computation.

The internal mesh series chosen can have an impact on the calculated value of <math>S_{20}</math> via the <math>P_{100}</math> parameter, which subsequently affects the computed mill load and discharge. As such, the internal mesh top size, <math>d_1</math>, is specified by the user. This ensures consistency when model parameters are transferred from once instance to another.

=== Multicomponent modelling ===

Published formulations of the Variable Rates AG/SAG model and its predecessors only consider feeds and loads consisting of a single ore type.{{Leung et al. (1987)}}{{Kojovic et al. (2012)}}{{Bueno et al. (2013)}}

In practice, mill feeds are often multicomponent, consisting of ores and minerals blended from multiple, differing sources. Recirculation of coarse, harder material for additional grinding passes adds to this complexity.

This implementation of the Variable Rates AG/SAG model addresses multicomponent feeds in a simple way:{{Bueno et al. (2013)}}

# Each ore type in a mill feed is assigned a separate set of density (<math>\rho_{\rm S}</math>) and hardness (<math>A</math>, <math>b</math>, <math>t_{\rm a}</math>) parameters.
# Ore masses-by-size are converted to volumes-by-size via the density parameters. 
# The <math>S_{20}</math> parameter value is determined on a volumetric basis for the combined ore types (and balls).
# The <math>(E_{\rm cs})_q</math> parameter value is determined using the overall density of all ore types in the mill load (i.e. the harmonic mean).
# Different appearance functions are then applied to separate population balance computations for each ore type in the feed. 

This multicomponent approach allows harder and coarser ore particles to accumulate in the mill load and preferentially concentrate in coarse recycle streams, as would be expected in practice.

Multiple ore types are excluded from the model formulations found in the previous sections of this article for clarity, but steps 1 - 5 above are automatically undertaken during model calculation. 

The multicomponent formulation reverts to the original single ore approach when only one ore type is present, or each ore type of a multicomponent feed is assigned the same values of <math>\rho_{\rm S}</math>, <math>A</math>, <math>b</math>, and <math>t_{\rm a}</math>.

=== Additional notes ===

==== Breakage rates and mill load ====

An important, and potentially overlooked, limitation of the Variable Rates AG/SAG mill model is the insensitivity of the breakage rate relationships to mill load. Mill simulations should therefore use mill loads close or equal to the load observed during model fitting, or 25% for design activities.{{Bailey et al. (2009)}}

==== Slurry pool ====

Various published descriptions of the Variable Rates AG/SAG mill suggest that slurry pooling phenomena are excluded from slurry hold-up and power draw estimations.{{Morrell and Morrison (1996)}}{{Kojovic et al. (2012)}}{{Bueno et al. (2013)}}

This implementation similarly excludes slurry pooling, and any model results returning values of <math>U > 1</math> should be inspected carefully.

== Excel ==

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

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

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>\mathit{Parameters} =
\begin{bmatrix}
D\text{ (m)}\\
L\text{ (m)}\\
D_{\rm t}\text{ (m)}\\
\alpha_{c}\text{ (deg.)}\\
\phi\text{ (frac)}\\
A_{\rm OF}\text{ (m}^{\text{2}}\text{)}\\
f_{\rm p}\text{ (m}^2\text{/m}^2\text{)}\\
x_{\rm p}\text{ (mm)}\\
x_{\rm g}\text{ (mm)}\\
x_{\rm m}\text{ (mm)}\\
\gamma\text{ (m/m)}\\
k\text{ (-)}\\
J_{\rm B}\text{ (v/v)}\\
\rho_{\rm B}\text{ (t/m}^3\text{)}\\
d_{\rm B,Top}\text{ (mm)}\\
F_{80}\text{ (mm)}\\
d_1\text{ (mm)}\\
\varepsilon\text{ (v/v)}\\
(Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\
\rho_{\rm L}\text{ (t/m}^{\text{3}}\text{)}\\
u\\
v\\
\end{bmatrix},\;\;\;\;\;\;

\mathit{Size} = \begin{bmatrix}
\hat{d}_{1}\text{ (mm)}\\ 
\vdots\\ 
\hat{d}_r\text{ (mm)}\\ 
\end{bmatrix},\;\;\;\;\;\;

\mathit{MillNewFeed} = \begin{bmatrix}
(Q_{\rm M,F})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,F})_{1m}\text{ (t/h)}\\ 
\vdots & \ddots & \vdots\\ 
(Q_{\rm M,F})_{r1}\text{ (t/h)} & \dots & (Q_{\rm M,F})_{rm}\text{ (t/h)}\\ 
\end{bmatrix},\;\;\;\;\;\;

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


:<math>
\mathit{BallSizing} = \begin{bmatrix}
(d_{\rm B})_1 \text{ (mm)} & (\mathit{MF}_{\rm B})_1\text{ (}%\text{ w/w)}\\
\vdots& \vdots\\
(d_{\rm B})_s \text{ (mm)} & (\mathit{MF}_{\rm B})_s\text{ (}%\text{ w/w)}\\
\end{bmatrix},\;\;\;\;\;\;

\mathit{RConst} = \begin{bmatrix}
\mathit{RConst}_1 \text{ (} \ln \rm h^{-1} \text{)}\\ 
\vdots\\ 
\mathit{RConst}_5 \text{ (} \ln \rm h^{-1} \text{)}\\ 
\end{bmatrix},\;\;\;\;\;\;

\mathit{OreBreakageParams} =
\begin{bmatrix}
A_1\text{ (}%\text{ w/w)} & \dots & A_m\text{ (}%\text{ w/w)}\\
b_1 & \dots & b_m\\
(t_{\rm a})_1\text{ (}%\text{ w/w)} & \dots & (t_{\rm a})_m\text{ (}%\text{ w/w)}\\
\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})_{r1}\text{ (t/h)} & \dots & (Q_{\rm M,R})_{rm}\text{ (t/h)}\\ 
\end{bmatrix}^*
</math>

where:
* <math>(Q_{\rm M,F})_{\rm L}</math> is the mass flow feed rate of liquids into the mill (t/h)
* <math>\rho_{\rm L}</math> is the Specific Gravity or density of liquids in the feed (- or t/m<sup>3</sup>)
* <math>u</math> is an index of the Appearance function to view in the results
* <math>v</math> is an index of the Appearance function to view in the results
* <math>m</math> is the number of ore types
* <math>r</math> is the number of intervals of the external mesh series
* <math>s</math> is the number of intervals of the ball mesh series below the top size, including the submesh
* <math>\hat{d}_i</math> is the size of the external square mesh interval that feed mass is retained on (mm)
* <math>\hat{d}_{i+1}<\hat{d}_i<\hat{d}_{i-1}</math>, i.e. descending size order from top size (<math>\hat{d}_{1}</math>) to sub mesh (<math>\hat{d}_{p}=0</math>)
* <math>(d_{\rm B})_i</math> is the size of the square mesh interval that balls are retained on (mm)
* <math>(\mathit{MF}_{\rm B})_i</math> is the mass fraction of balls retained on ball mesh series interval <math>i</math> (% w/w)
* <math>^*</math> indicates the <math>\mathit{MillRecycleFeed}</math> 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:

:<math>
\mathit{mdUnit\_AGSAG\_VariableRates} = 
\begin{bmatrix}

\begin{bmatrix}
\text{Iterations}\\
\text{Iteration error}\\
V\text{ (m}^{\text{3}}\text{)}\\
\text{Mill speed (rpm)}\\
Q_{\rm V,F}\\
M_{\rm S}\text{ (t)}\\
M_{\rm L}\text{ (t)}\\
M_{\rm B}\text{ (t)}\\
M\text{ (t)}\\
J_{\rm t}\text{ (v/v)}\\
U\text{ (v/v)}\\
k_{\rm g}\text{ (-)}\\
m_1\text{ (-)}\\
m_2\text{ (-)}\\
d_{\rm max}\text{ (h}^{-1}\text{)}\\
S_{20}\text{ (mm)}\\
E_q\text{ (kWh)}\\
\rho_{\rm c}\text{ (t/m}^3\text{)}\\
\theta_{\rm S}\text{ (rad)}\\
\theta_{\rm T}\text{ (rad)}\\
r_{\rm i}\text{ (m)}\\
P_{\rm NoLoad}\text{ (kW)}\\
P_{\rm Net}\text{ (kW)}\\
P_{\rm Gross}\text{ (kW)}\\
\end{bmatrix}

&

\begin{array}{cccccccc}

\begin{bmatrix}
\hat{d}_1\text{ (mm)}\\
\vdots\\
\hat{d}_q\text{ (mm)}
\end{bmatrix}

&

\begin{bmatrix}
(Q_{\rm M,P})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,P})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,P})_{q1}\text{ (t/h)} & \dots & (Q_{\rm M,P})_{qm}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{bmatrix}
(M_{\rm S})_{11}\text{ (t)} & \dots & (M_{\rm S})_{1m}\text{ (t)}\\
\vdots & \ddots & \vdots\\
(M_{\rm S})_{q1}\text{ (t)} & \dots & (M_{\rm S})_{qm}\text{ (t)}\\
\end{bmatrix}

& 

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

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

&

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

&

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

&

\begin{bmatrix}
(E_{\rm cs})_{1}\text{ (kWh/t)}\\
\vdots\\
(E_{\rm cs})_n\text{ (kWh/t)}\\
\end{bmatrix}

&

\begin{bmatrix}
A_{u1}\text{ (frac)}\\
\vdots\\
A_{un}\text{ (frac)}\\
\end{bmatrix}

&

\begin{bmatrix}
A_{v1}\text{ (frac)}\\
\vdots\\
A_{vn}\text{ (frac)}\\
\end{bmatrix}\\

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

\end{array}

\end{bmatrix}
</math>

where:
* <math>\text{Iterations}</math> is the number of internal computation steps required to converge the load
* <math>\text{Iteration error}</math> is the numerical error of the converged load approximation
* <math>Q_{\rm V,F}</math> is the flow rate of pulp into the mill (m<sup>3</sup>/h)
* <math>\text{Mill speed}</math> is the [[Tumbling Mill (Speed)|rotational rate of the mill]] (rpm)
* <math>M_{\rm S}</math> is the mass of ore solids in the mill (t)
* <math>M_{\rm L}</math> is the mass of liquids in the mill (t)
* <math>M_{\rm B}</math> is the mass of balls in the mill (t)
* <math>M</math> is the total mass of ore, liquids and balls in the mill (t)
* <math>Q_{\rm M,P}</math> is product mass flow rate (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:AGSAGVariableRates1.png|left|frame|Figure 8. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]] 
| [[File:AGSAGVariableRates2.png|left|frame|Figure 9. Example showing the selection of the '''Size''' (dark red frame), '''OreSG''' (green frame), '''MillNewFeed''' (purple frame) and '''MillRecycleFeed''' (light red frame) arrays in Excel.]] 
|}
{| 
|- style="vertical-align:top;"
| [[File:AGSAGVariableRates3.png|left|frame|Figure 10. Example showing the selection of the '''BallSizing''' (purple frame), '''RConst''' (brown frame), and '''OreBreakageParams''' (teal frame) arrays in Excel.]] 
| [[File:AGSAGVariableRates4.png|left|frame|Figure 11. 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.

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

==== Mill page ====

The Mill page is used to specify the input parameters for the mill model.

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''VariableRates''

{{SysCAD (Text, Help Link)}}

|-
! colspan="3" style="text-align:left;" |''Requirements''
{{SysCAD (Text, NumParallelUnits)}}
|-
! colspan="3" style="text-align:left;" |''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.
|-
! colspan="3" style="text-align:left;" |''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.
|-
! colspan="3" style="text-align:left;" |''Ball''
|-
|BallLoad
|Input
|Ball load fraction.
|-
|BallSG
|Input
|Density (Specific Gravity) of ball media.
|-
|NumBallMeshSizes
|Input
|Number of ball mesh sizes below the top size, including the submesh.
|-
|BallTopSize
|Input
|Top size of new ball media.
|-
|Size
|Input / Display
|Ball sizing intervals.
|-
|Load
|Input
|Mass fraction retained of ball media in each ball sizing interval.
|-
! colspan="3" style="text-align:left;" |''RFunction''
|-
| RSize
|style="background: #eaecf0" | Display
|Spline knot positions.
|-
| RConst
|Input
|Values of <math>\mathit{RConst}</math> at each spline knot position.
|-
! colspan="3" style="text-align:left;" |''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.

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''Ore''
|- style="vertical-align:top;"
|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.
|-
|colspan="3" | [[File:AGSAGVariableRates5.png]] [[File:AGSAGVariableRates6.png]]
|}

==== Results page ====

This page is used to display the model results.

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''Results''
|-
! colspan="3" style="text-align:left;" |''Solver''
|- 
|Iterations
|style="background: #eaecf0" | Display
|Number of iterations to converge internal load solver.
|- 
|IterationError
|style="background: #eaecf0" | Display
|Numerical approximation error of internal load solver.
|-
! colspan="3" style="text-align:left;" |''Mill Properties''
|- 
|MillVolume
|style="background: #eaecf0" | Display
|Internal volume of the mill.
|- 
|MillSpeed
|style="background: #eaecf0" | Display
|Rotational speed of the mill.
|- 
|MillFeedRate / Feed.SLQv
|style="background: #eaecf0" | Display
|Volumetric feed rate of pulp into the mill.
|-
! colspan="3" style="text-align:left;" |''Mill Contents''
|- 
|OreMass
|style="background: #eaecf0" | Display
|Mass of ore (solids with PSD) in the mill.
|- 
|LiquidMass
|style="background: #eaecf0" | Display
|Mass of liquids in the mill.
|- 
|BallMass
|style="background: #eaecf0" | Display
|Mass of ball media in the mill.
|- 
|TotalChargeMass
|style="background: #eaecf0" | Display
|Total mass of ore, liquids and balls in the mill.
|- 
|VolTotalLoad
|style="background: #eaecf0" | Display
|Volumetric fraction of mill volume of total charge (ore, liquids, balls and void space).
|-
! colspan="3" style="text-align:left;" |''Mill Discharge''
|- 
|m1
|style="background: #eaecf0" | Display
|Parameter of the Austin mill holdup relationship.
|- 
|m2
|style="background: #eaecf0" | Display
|Parameter of the Austin mill holdup relationship.
|- 
|dMax
|style="background: #eaecf0" | Display
|Maximum discharge rate of load volume through the grate.
|-
! colspan="3" style="text-align:left;" |''Charge Properties''
|- 
|S20
|style="background: #eaecf0" | Display
|Size of the top (largest) 20% of the load.
|- 
|ChargeDensity
|style="background: #eaecf0" | Display
|Density of the charge.
|- 
|U
|style="background: #eaecf0" | Display
|Fraction of charge void space filled with slurry.
|- 
|ThetaShoulder
|style="background: #eaecf0" | Display
|Angular position of the charge shoulder.
|- 
|ThetaTue
|style="background: #eaecf0" | Display
|Angular position of the charge toe.
|- 
|ChargeSurfaceRadius
|style="background: #eaecf0" | Display
|Radius of the inner charge surface.
|-
! colspan="3" style="text-align:left;" |''Power''
|- 
|NoLoadPower
|style="background: #eaecf0" | Display
|No-load power draw of the mill.
|- 
|NetPower
|style="background: #eaecf0" | Display
|Net power draw of the mill.
|- 
|GrossPower
|style="background: #eaecf0" | 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.

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''Rates''
|-
|Size
|style="background: #eaecf0" | Display
|Size of each interval in internal mesh series.
|-
|MeanSize
|style="background: #eaecf0" | Display
|Geometric mean size of each interval in internal mesh series.
|-
| R
|style="background: #eaecf0" | Display
| Value of breakage rate, <math>R_i</math>, for each size interval, for each ore species. 
|-
| D
|style="background: #eaecf0"; | Display
| Value of discharge rate, <math>D_i</math>, for each size interval. 
|-
| Ecs
|style="background: #eaecf0"; | 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.

{{SysCAD_Table_Header}}

{{SysCAD (Text, Distribution)}}

|-
! colspan="3" style="text-align:left;" |''Load''
|-
|SolidMass / SMt
|style="background: #eaecf0" | Display
|The mass of solids with the SysCAD size distribution property currently in the mill.
|-
|LiquidMass / LMt
|style="background: #eaecf0" | Display
|The mass of liquids plus solids without the SysCAD size distribution property currently in the mill.
|-
|BallMass / BMt
|style="background: #eaecf0" | Display
|The mass of ball media in the mill.
|-
|Size
|style="background: #eaecf0" | Display
|Size of each interval in the external mesh series.
|-
|MeanSize
|style="background: #eaecf0" | Display
|Geometric mean size of each interval in the external mesh series.
|-
|Load
|style="background: #eaecf0" | Display
|The mass of solids with the SysCAD size distribution property currently in the mill, by size and species.
|}

{{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={{SysCAD (Text, UnitType Prefix)}}Mill}}

{{SysCAD (Page, About)}}

== See also ==

* [[Tumbling Mill (Slurry Flow)]]
* [[Tumbling Mill (Power, Morrell Continuum)]]
* [[Crusher (Whiten)]]

== References ==

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