What to do when Bond fails?

I had an interesting question from one LinkedIn fan that is worth a public discussion. The background is the person is working with a new dry grinding apparatus and needs to generate a model to size the industrial machine based on laboratory results. "From what I’ve done, regular Bond and Kick models aren’t very effective as the basis of this model". This is not unusual, and we have other tools to work with situations when the Bond Third Theory model fails.

Reviewing the theory of specific energy models

But first, let's back up a step – the Bond equation (more specifically, the Third Theory equation) is used to predict the specific energy E given a coefficient 10×Wi and two sizes, the feed size F?? and a product size (sometimes T??, though it is more common to use P?? for the product). There is an exponent of -? on the size terms, which is another way of writing "one over the square root". We'll use sizes in μm in the discussion for compatibility with industry practice.

Bond's model is part of a family of models that are derived from Hukki's Conjecture that most specific energy models are forms of a differential equation:

where: x is particle size, K is some constant coefficient and -f(x) is a variable exponent that applies to the particle size. A rearranged form of this equation looks like this:

If one assumes that the exponent term is reasonably constant over the size range being integrated, then the messy exponent gets replaced with a constant and we have a relative easy first-order integration where the constant exponent and the constant coefficient (K) results in another constant coefficient, let that be C, and a new constant exponent, let that be α. This is equivalent to the integrated form of Charles Equation from 1957.

It should be obvious that Bond's Third Theory equation is the case where α=–? and C=10Wi, but there are lot of other equations that fit this form that are used in different situations. The Rittinger model (from 1867) has α=–1:

A simplifying assumption is possible for large reduction ratios where F?? is significantly greater than P??, the feed term becomes negligible because "one over a large number" is a small value. This allows a different model, the signature plot that neglects the feed entirely (see my video on fine grinding for more on this model https://youtu.be/ytvcJX-hBKE).

The signature plot has two measured parameters, the coefficient C and the exponent -b. These parameters are measured in the lab as a part of an Isamill test (used for sizing fine-grinding Isamills) or from a Levin grindability test using a Bond ball mill apparatus.

Recommendation

This signature plot model is the recommended form for the new model that answers the question of what to do when Bond and other models fail. It is likely that the new grinding apparatus and the size range being investigated do not generate a constant exponent on the size term, so it needs to be measured.

Important note – the E term is the specific energy consumption of the industrial machine and not the specific energy consumption of the laboratory apparatus. According to Mike Daniel 's PhD thesis, the specific energy consumption of the Bond apparatus is about two-thirds of the specific energy consumption of a "standard" 8-ft diameter by 8-ft long industrial ball mill. Bond generated an empirical calibration equation to convert the laboratory apparatus results to match the observed industrial machine behaviour:

There are a few laboratory apparatus where the specific energy consumption of the laboratory apparatus match the specific energy consumption of the industrial machine – the IsaMill Ultrafine Grinding mentioned earlier is an example – which makes the mill sizing easy when you have laboratory results. Most grinding machines don't have this property and require empirical calibrations between laboratory results and industrial machine specific energy consumption. Examples of this are the three Bond laboratory apparatus (ball mill, rod mill, and crushers), SAG?Grindability Index (SGI, equivalent to SPI), SAGDesign (the empirical conversion from mill revolutions to E is done automatically by Starkey & Associates Inc-Consulting Engineers ), and sizing of Metso Vertimills from a jar mill test.

Proposed Method

1. Collect industrial machine specific energy consumption values and laboratory samples corresponding to the industrial survey data.
2. Check that the slope of the Particle Size Distributions (PSDs) from the industrial machine are reasonably consistent in slope when grinding finer. Use the Granulometrics web tool to check the slopes, hopefully the PSD fits the Gaudin-Schuhmann model where the slope is the exponent (https://granulometrics.com). If this slope is consistent, then Specific Energy Models are suitable as one can use a single point on the PSD (like the 80th percentile) to represent the whole PSD. If not, then none of this will work and a much more complicated Population Balance Model technique must be used, instead.
3. Generate laboratory results using the survey samples.
4. Look for industrial samples with the same feed hardness and different product sizes. Use these surveys to figure out what the exponent -b of your signature plot should be for that feed.
5. Repeat this for different feeds to determine if -b is constant for all feed hardness, or if it varies. It is desirable for the exponent to be constant, if possible, as it makes the model simple (Bond's Third Theory is an example of a constant exponent model).
6. If the exponent is constant, then fix that value and determine how the coefficient varies with the laboratory test results.
7. If the exponent is variable, then both the exponent and the coefficient must be simultaneously be determined from the laboratory test results.

This procedure is based on the assumption that the feed and product sizes are sufficiently different that the feed can be neglected. If this assumption isn't valid, then a much more difficult model fit is required to the Charles Equation.

The model doesn't need to be perfect, it needs to be "good enough" to get within "one mill size" of the perfect choice. Laboratory grindability results are usually uncertain within ±5% to ±8%, so don't waste time trying to calibrate a model that works down to the third decimal place.

ALL MODELS ARE WRONG, BUT SOME ARE USEFUL.

Further reading

Here are some other models used in minerals comminution that follow the form of the Signature Plot, such as the Peter Amelunxen 's SGI?model (equivalent to the CEET1 model using SPI) where the constant is (5.9 × SGI?·??)×CFsag and the product is the transfer size from the SAG?mill to the ball mill (T??) that has an exponent of -0.275.

There are also more complicated models that fit the Hukki Conjecture, such as Morrell's Mi model:

where Stephen M. derived a variable exponent f(x) = -0.295 - 10?? x. This exponent is clearly incorrect because the boundary condition of fine grinding (exponent for x<74 μm tends towards -0.295) doesn't match the signature plot exponents (between -1.5 and -3.0 depending on the material). This doesn't matter over short size ranges, but a separate correction to the Mib value is done if the laboratory P?? deviates significantly from the industrial machine P??.

The Kick model, sometimes used to describing blasting, has different exponents for the feed (-1) and product (1). It is sometimes shown in literature with a logarithm, but Kick's text describes size reduction as a ratio, so let's go with this:

Numerous other models exist that don't use the 80th percentile, focus on the feed size instead of the product size, and many other variations that may or may not fit Hukki's Conjecture. As mentioned earlier, any empirically calibrated model that gets you a close enough answer is better than mathematical perfection cursed by an unwieldy model.

I?hope folks find this useful.

.:. Alex Doll