Observations on signature plots from Bond BM Wi tests

I am working with some Bond ball mill grindability tests performed at a commercial laboratory. The lab ran three tests on the sample, each test at a different closing size. With this, we can observe the variation in work index (or Morrell's Mib value) with grind size and generate two signature plots (one based on Bond's work index, other based on Stephen M. 's Mi model). This is useful when one needs to adjust the P80 basis of work index tests because the flotation people keep changing their minds what the grind size should be.

First observation is that if you plot the "raw" specific energy consumption to go from the test feed F80 size to the three product P80 sizes, as in Figure 1. This is identical to how a signature plot appears from a laboratory stirred mill test, the feed size is assumed to have zero specific energy consumption.

There is a steady increase in specific energy consumption as one grinds finer; this is the expected outcome. The specific energy computed from work index is always a bit less than the specific energy consumption computed from Mib value. This is normal, the two methods are calibrated to different data sets so will have different specific energy predictions.

The exponents on the regression equations probably look odd to people – it is reasonable to expect that since the size term of Bond's Third Theory is one over the square root (which is equal to an exponent of -½ in a signature plot), we expect to see an exponent of -0.5, and not -0.58. Similarly, the Mi specific energy equation (in its raw form) has an exponent that tends to -0.295 at small values, not -0.72 as we observe. ¿Qué pasa?

First thing is that the size range being observed in these tests is truncated with zero specific energy appearing at 2.4 mm – but both of these models expect the zero specific energy to be at infinite size for the exponent to match these expectations. So can we correct the signature plot for that and change the "zero basis?". Sure, we just add another point for the Ball Mill test feed size and assign the specific energy consumption for that point. We also escalate the computed specific energy consumptions for the other points by this zero-basis correction. In tabular form, the correction looks like this:

The generation of E_Wi and E_Mi at 2,389 µm is, unfortunately, a bit arbitrary because we have no grindability determination at that size class. The Bond value is generated using the Wi of the coarsest test, and the Mi value makes use of the Mib_target from Mib_ref equation published in the GMG guideline for Morrell circuit efficiency. The next step is a bit sensitive to this adjustment, so we note this risk and carry on.

With these corrected values, we can generate a new signature plot that is corrected to include the intrinsic specific energy at the feed size to the tests.

These two curves now look a lot closer to Bond's empirical exponent of -½, and (hilariously) it is Morrell's equation that is the closest of the two.

This leads to the second reason why exponents don't match the two models:  the models are empirically fit to a data set (see my post from yesterday https://www.linkedin.com/posts/alex-doll-66b57465_workindex-activity-7138416036315369472-vpp0). Any ore won't perfectly match the models, there is always some noise and/or error between what an ore does and what the models expect. Some of this error is correctable (the aforementioned Mib_target from Mib_ref is one such correction), but there is also some random noise that we need to live with. Know that it is there and that the farther one pushes models outside of their calibration range, the less certain the predictions will be.

Summary

• Signature plots can be generated from multiple Bond ball mill grindability tests conducted on duplicate samples. Signature plots aren't limited to stirred mills.
• The zero-basis of a signature plot is important. You get very different equations if the feed size to the test changes, and this can lead to design failures if the plant has a much coarser feed to a mill than the feed size the signature plot was generated at.
• There is a way to adjust the zero-basis size of a signature plot, although the adjustment is sensitive to how you assign the intrinsic specific energy of the feed sample. This can be useful when you need to shift a signature plot to a new size basis.
• The exponents that one observes in the signature plots rarely match the exponents built into the commonly used models. My opinion is that the exponent is an inherent property of the rock that can be measured. Exponents will usually be close to the model expectations, but there are always exceptional ores that don't fit the models.
• Avoiding these sorts of corrections and worrying about exponents is why it is always desirable to run a ball mill grindability test so that the P80 size is close to the desired product size of the industrial plant.
• Feasibility and design work should always measure the variation in work index (or exponent, or Mib) as a function of size. This allows future engineers to rescue the plant design when the grinding circuit product size target changes later on in the project. Typically measure this once on each major rock type (so only do these triplicate tests on two or three composite samples).