Category:Models

Models

This are two principal types of models in SAGMILLING.COM:

• Specific energy consumption models are used to estimate the amount of grinding energy required to grind a particular ore, and what throughput can be achieved by passing that ore through a particular set of grinding mills.
• Mill power draw models are used to estimate how much grinding energy can be generated by a particular mill geometry and charge.

The combination of a specific energy model of an ore's grindability with a "circuit" containing one or more mill power draw models permits the throughput of the particular ore in a particular grinding circuit to be simulated. In the most generalized form, the equations fit together like this:

• Specific energy (kW h/t) = mill power (kW) / throughput (t/h)

Two ways these models can be configured for a given ore (specific energy consumption is a property of that ore) are:

1. Knowing the desired throughput allows the circuit mill power to be calculated: kW = kW h/t * t/h
2. Knowing the circuit power draw allows the throughput to be calculated: t/h = kW / (kW h/t)

History of Power-based Specific Energy Consumption Circuit Models

A systematic relationship between energy consumed in grinding and size reduction was proposed by Peter von Rittinger in 1867, yielding the first Power Model and beginning of comminution as a science ^{Lynch & Rowland, 2005}. This was followed in 1885 by the theory of Friedrich Kick who proposed a much different power model. Instruments of that age did not allow accurate enough measurements to test the theories ^{Lynch & Rowland, 2005}, so the difference between the models was largely of academic interest. Not until instruments for motor power measurement and laboratory grinding mills were standardized in the late 1930's to 1940's were attempts made to bench-mark the theories of von Rittinger and Kick to operating grinding plants (one such attempt was Myers, Michaelson & Bond, AIME 1947. The failure of these rod mill and ball mill bench-mark studies to validate either model led to the development of a "Third Theory" by Fred Bond ^{Bond 1952}.

R. Hukki in 1962 showed that all three theories may be special cases of a spectrum of a larger, more generalized, theory where Kick best characterized the coarsest particles (blasting), Bond the middle size ranges (rod and ball milling), and von Rittinger the finest size classes (fine and ultra-fine grinding).

A later model by Stephen Morrell attempted to smoothly merge all three of the earlier models into a single continuum that was fitted to surveys Morrell conducted in Australasia and Africa. However, the exponents derived by Morrell are very different from the exponents predicted by Hukki and, in the finest size classes where the exponent tends to 0.30, do not match observed signature plots from tests involving stirred mills (e.g. Isamills and Vertimills where the exponent ranges between 0.5 and 3.0).

Theory of Power-based Specific Energy Consumption Models

Laboratory measurements of the grinding energy required to break rocks are input into emperical formulae, such as Bond's equations, the SGI/SPI equations or the "Mi" equations that use results from SMC tests. Different size classes are tested and, in general, the specific energy consumption models use the laboratory parameters to influence the similar size range in the model.

Taggart noted (and Bond adopted) several properties of common size reduction processes that allows modelling as a process of energy consumption by and ore^1,2. The underlying observations (and assumptions) of Power Models are:

• Any particle size distribution can be modelled by a single value. Generally, it is the 80% passing size that is used as being indicative of the entire size distribution.
• Particle size distributions of "normal" size reduction processes produce "similar" particle size curves, generally with the feed and product having the same proportions of "coarse" and "fines" where only the sizes that define coarse and fine are shifted.

Mill Power Draw Models

Mill power models predict the amount of power (measured at the mill shell) that will be drawn by a mill of a particular geometry under particular operating conditions. A variety of models are available for a particular type of mill, and all should give similar answers for most conditions. If mill models give very different power draw predictions, then carefully check that the operating conditions entered are valid and not far outside the calibration range of the models.

In general, the mill power draw models are independent of the ore being processed (ore density matters, but feed and product sizes should not affect power draw if the mill is run properly). The specific energy model uses the ore grindability test results to estimate energy requirements to achieve a particular size reduction, then it calls on the mill power draw model to determine how much energy is actually available. Dividing the energy available by the specific energy consumption results in the throughput.