Model:Bond/Barratt SABC Models

Bond/Barratt Specific Energy Consumption SABC Model

This is a SAG or AG mill plus ball mill model that estimates the overall circuit specific energy consumption using the classical Bond work index equation for multi-stage crushing and single-stage ball milling (E_ssbm) including Rowland efficiency factors. The circuit E_total is equal to the E_ssbm plus an inefficiency factor (CF) related to the difference in grinding efficiency of the two types of circuits. The SAG mill specific energy consumption (E_SAG) is calculated using the 1979 Barratt equation and the ball mill specific energy consumption (E_bm)

This model includes a phantom cyclone effect in the equations due to the ball mill being calculated by difference and not being calculated by the normal Bond equation. The resultant operating work index of the ball mill will vary according to the ratio of the ball mill, rod mill and crushing work index values and is in the range of 80% of the measured ball mill work index value.

Testwork Required

• Bond ball mill work index
• Bond rod mill work index
• Bond crushing (low-energy impact) work index

Formulae

$E_{ssbm} = Wi_{C} \times \left ( \tfrac {10}{\sqrt{ 9400 }} - \tfrac {10}{\sqrt{ F_{80} }} \right ) + Wi_{RM} \times \left ( \tfrac {10}{\sqrt{ 2100 }} - \tfrac {10}{\sqrt{ 9400 }} \right )\times EF_4^{RM} + Wi_{BM} \times \left ( \tfrac {10}{\sqrt{ P_{80} }} - \tfrac {10}{\sqrt{ 2100 }} \right ) \times EF_4^{BM} \times EF_5$ ^{Rowland, 2006}

$E_{total} = E_{ssbm} \times (1+CF)$

$\begin{align} E_{SAG} = \Big[ & Wi_{C} \times \left ( \tfrac {10}{\sqrt{ P_{C} }} - \tfrac {10}{\sqrt{ F_{80} }} \right ) + Wi_{RM} \times \left ( \tfrac {10}{\sqrt{ P_{R} }} - \tfrac {10}{\sqrt{ P_{C} }} \right ) \times EF_4^{RM} \\ & + Wi_{BM} \times \left ( \tfrac {10}{\sqrt{110} } - \tfrac {10}{\sqrt{ P_{R} }} \right ) \times EF_4^{BM} \Big] \times 1.25 - Wi_{BM} \times \left ( \tfrac {10}{\sqrt{ {110} }} - \tfrac {10}{\sqrt{ T_{80} }} \right ) \end{align}$ ^{Barratt, 1989}

$E_{bm}= E_{total} - E_{sag}$

As with most models, the power split between the SAG and ball mills is evaluated and the transfer size (T₈₀) necessary to balance the power draw between the two stages is estimated. If that transfer size is within the specified limits, then the calculation proceeds to calculate the E_SAG and E_bm using that transfer size. If the transfer size to balance the power is outside the specified limits, then the transfer size size is forced to the limit and the E_SAG and E_bm values are then calculated. SAG-limited circuits (where the T₈₀ hits the upper limit) will result in the ball mill over-grinding unless the ball mill speed is reduced to control the grind size (in the case of a variable speed mill). Ball-limited circuits (where the T₈₀ hits the lower limit) will result in the charge level in the SAG mill dropping until its power draw matches the throughput limit dictated by the ball mill. It is important to realize that once the SAG mill charge level drops to be equal to the ball charge, the model will be operating "out of range" and the predictions will not be valid.

Circuits that operate at or near the transfer size limits should be less efficient than the default formulas will estimate. If you have a design that is consistently operating around the limits, then consider adding a contingency to inflate the specific energy consumption estimates to account for this poor grinding efficiency. Alternatively, if you are designing a new circuit then choose a different combination of SAG and ball mills that operates in a more reasonable transfer size range.

The optimum feed size for the rod mill oversize feed factor (EF₄^RM) in the Barratt E_SAG equation is calculated using the greater of the sample's rod mill or crushing work index.

$F_{O}= 16000 \times \sqrt{\frac{13}{Max(Wi_{RM}, Wi_{C}) \div 1.1023} }$

The EF₄^RM is calculated greater of the crushing work index or the rod mill work index.

$EF_{4}^{RM} = \frac{RR+ \bigl(\frac{Max(Wi_C, Wi_{RM})}{1.1023} -7 \bigr) \times \frac {F_{80}-F_{O}}{F_{O}}}{RR}$

The ball mill oversize feed factor (EF₄^BM) is always calculated with the rod mill work index for the optimum feed size and the ball mill work index for EF₄^BM.

$F_{O}= 4000 \times \sqrt{\frac{13}{Wi_{RM} \div 1.1023} }$

$EF_{4}^{BM} = \frac{RR+ \bigl( \frac{Wi_{BM}}{1.1023} -7 \bigr) \times \frac {F_{80}-F_{O}}{F_{O}}}{RR}$

The model assumes that the calculated value of E_ssbm + the specified contingency is greater than the sum of E_sag + E_peb; otherwise E_bm will be negative. The model does a check to see if E_bm calculated is negative (or close to zero) and, if so, will substitute the Raw Bond Model equation estimate for E_bm.

Recommended Usage

This model should be the lower end-member of the range of Bond/Barratt style models. It is useful when interpreting pilot plant results to determine where the pilot plant results fall relative to the spectrum of models. It can be used as a realistic model for design work in softer ore types like copper porphyries. The Optimized Bond/Barratt model is best used for low competency ores, such as those found in the Western Cordillera of North America or the Andes, where the impact crushing work index is similar to, or less than the rod mill and ball mill work index values. These ores tend to demonstrate a phantom cyclone effect.

Discussion, pebble crushing

The pebble crusher is assumed to be part of the SAG mill for the purposes of this calculation. E_peb is calculated as the power drawn by the pebble crusher divided by the circuit throughput. This E_peb is then deducted from E_sag.

The total circuit specific energy consumption is generally higher when no pebble crusher is present in a circuit. The rule of thumb is apply the CF value of 0.15 for SAB circuits (5% higher overall specific energy consumption) ^{Bennett, Dobby & Kosick, SAG 2001}. Published data from Cadia confirms a 5% difference in throughput (and therefore, specific energy) when the pebble crusher is fully utilized.^{Hart, Valery, Clements, Reed, Song & Dunne, SAG 2001}.

Discussion, Rowland EF factors

The model is only interested in the EF factors that are properties of the rock, namely the coarse feed EF4 factor and the fine product EF5 factor. The EF factors that correspond to machine characteristics are not included (EF₁, EF₂, EF₃, EF₆, EF₇ and EF₈) as they are intended to capture the inefficiency of certain types of milling circuits. The assumption in this model is that the E_ssbm is the "maximum efficiency" possible for a grinding circuit, to which we apply an empirical adjustment for SAG milling (Cf) which encapsulates any other EF factors in a form suitable for a SAG process.

Model defaults

• Maximum T₈₀, µm = 6000
• Minimum T₈₀, µm = 400
• P_C for Essbm= 9400
• P_C for Easag= 18850
• P_R= 2100
• CF = 0.10