Many types of mathematical models are used in comminution and mineral processing.  These models can generally be classified into one or more of the four categories shown above.

Mill Power Draw Models

The purpose of a mill power draw model is to predict what the mill power and/or torque draw will be for a mill of a certain geometry containing a specified charge being tumbled at a specified rotation rate.  These models are used be designers to select mill and motor combinations for new plant designs, and are used by operators to “benchmark” the operation of an existing mill.  An operating mill that draws power equal to or exceeding the model is generally a healthy mill.

Examples used in industry (and example software I am aware of):

• Morrell C-model (JK SimMet & SAGMILLING.COM) and E-model
• Austin SAG model (SAGMILLING.COM)
• Hogg & Fuerstenau (MolyCop Tools)
• Nordberg model (SAGDesign and SAGMILLING.COM)

Specific Energy Models

The purpose of specific energy models is to predict how much grinding energy is required to break a particular rock down to a specified size.  A single size category, typically 80% passing, is used in order to simplify the equations.  These models are also used in plant design as they dictate how much grinding power is required to break a rock at a desired throughput rate.  Specific energy models are combined with mill power draw models in almost all mill designs.

Benchmarking the grindability of an operating plant against specific energy models is a useful guide to validating that the plant is operating efficiently.  Ore grindability has a big influence on the energy consumption and throughput; only by measuring the ore can one assess the plant performance.

Examples of specific energy models (and the associated laboratory tests):

• Bond models (work index tests for ball milling, rod milling and crushing)
• SAGDesign (SAGDesign and Sd-BWI)
• SPI and SGI models (SAG Grindability Index and SAG Power Index, ball Wi)
• Morrell Mi (SMC Test and Mib from standard Bond ball mill work index)

Population Balance Models

For situations where knowledge of the whole size distribution is required, more complex population balance models are used.  These involve considering the whole size distribution being fed to a mill and in the mill product, along with math that describes the breakage and classification of each size class.  The math is much more complex than the Specific Energy Models, but Population Balance Models can be used in situations that confuse the single size class method – such as scalped feed or pebble load predictions.

The language used to describe the math varies a bit between the different varieties of population models, but they generally all have a (matrix) term that describes how frequently each size class breaks and another (matrix) term that describes the daughter product sizes generated from each breakage.  These two terms are usually empirically fit to mill surveys due to the quantity of unknown parameters that must be specified.

Examples of Population Balance Models are those by Herbst & Fuerstenau, Whiten, and Austin.  Examples of software that includes Population Balance Models are JK SimMet and MolyCop Tools.

There is some overlap between Population Balance models and Specific Energy Models, with the CEET2 model by SGS containing a specific energy breakage model combined with a population balance ability to predict size distributions.

Discrete Element Modelling

The newest and most computationally intensive models to appear in the Industry are the Discrete Element Models (DEM) where fundamental Newtonian physics is used to simulate the motion of particles in industrial settings, such as particles moving inside a grinding mill or cascading off a conveyor into a chute.  These models are widely used to select mill lifter geometry for maximum grinding efficiency and to design chutes so as to minimize liner wear.

The DEM models are being actively extended by a number of authors.  One intriguing application is the marriage of DEM with Computational Fluid Dynamics (CFD) for use in simulating and optimizing SAG mill discharge grates and pulp discharge lifter systems.  Grate function is notoriously difficult to predict with current tools meaning that operators have to experiment with different grate percentage open area and slot geometry to find the best combination for their needs.  DEM & CFD, once they are fully combined and calibrated, offer a way to determine the best grate geometry even before a mill is installed.

There is also overlap of DEM with Population Balance Models where authors are combining breakage mathematics into element collisions.  A simulated impact with balls or other components of the mill system can result in a rock’s breakage into a series of smaller particles.  The benefit of this approach is to avoid the need for empirical breakage rate measurements in Population Balance Models, making them more useful in design situations where mill surveys aren’t possible and the empirical measurements must be adapted from other mills.

DEM models require a lot of calibration and experience to operate, and it is not wise for casual users to try operating them.  I won’t be naming software for this class of models in order to protect the do-it-yourselfers from themselves.

Conclusion

Mineral comminution makes use of a lot of different types of mathematical models.  Generally the right model for a particular job is the simplest model that gets a “good enough” answer.  The industry has invented a number of models that do effectively the same job, differing mostly in their calibration or the specific class of equipment being simulated.  Usually two equivalent models should give “close enough” answers – if they differ greatly, then that is a warning that extra care is needed before continuing the investigation.