# Austin SAG model

## History

The SAG mill model by proposed by Leonard Austin (1990) was largely based on modifications of earlier tumbling mill models by Hogg & Fuerstenau and F. Bond. The model uses a kinetic-potential energy balance to describe the power draw of a mill charge. Many geometric components of the earlier models were fit to empirical relationships measured by Austin.

## Model Form

The power draw model for a SAG mill cylinder of the following form:

Where:

*P*is the power evolved at the mill shell, kW*K*and*A*are empirical fitting factors (use 10.6 and 1.03, respectively)*D*is the mill effective diameter (inside the effective liner thickness), m*L*is the mill effective grinding length (also referred to as the 'belly length), m*J*is the mill total volumetric filling as a fraction (eg. 0.30 for 30%)_{total}*ε*is the porosity of the rock and ball load (use 0.3)_{B}*w*is the charge %solids, fraction by weight (use 0.80_{C}^{Doll, 2013})*J*is the mill volumetric filling of balls as a fraction (eg. 0.10 for 10%)_{balls}*ρ*is the density of component_{X}*X*, t/m^{3}*φ*is the mill speed as a fraction of critical (eg. 0.75 for 75% of critical)_{C}

### Mill cone angles

The model supports flat-ended mills (cone angle of 0°) or a cone angle of 15°. Any other value entered for the cone angle will be treated as 15°.

To account for cone ends of mills, an allowance of 5% is used ^{Doll, 2013} instead of the formula proposed by Austin.

### Percent solids

The model is run with a fixed 80% solids by weight ^{Doll, 2013}. This is because the form of the equation proposed by Austin appears to have the %solids term in the wrong place (denominator of the expression rather than the numerator where other models put it). The calibration presented at IMPC 2016 suggests the fixed %solids term is reasonably valid over the range of 60% to 80% solids.

## Model Inputs

**the Austin model** requests the following mill data:

- mill nominal diameter inside shell
- mill effective grinding length
- ball load as vol percentage (eg. 10)
- ball density
- total charge load as vol percentage (eg. 30)
- pulp percent solids
- mill speed
- liner effective thickness
- cone end angle (only values 0-5 or >5 matter, minor angle changes have no effect)

**Motor characteristics** are also requested:

- Motor rated power (output shaft)
- Mechanical efficiency of downstream drive (pinions, gearboxes)
- Motor efficiency and any other efficiency factors to the DCS measurement position in the network.
- Motor rated speed, in units of mill RPM (not motor RPM, must multiply by gear ratios).
- Quantity of pinions (and motors)