Empirical conversions, SPI? & SGI
TLDR version (revised 2020-04-14):
- (SPI/SGI, minutes) = 0.89 × (Bond Wi_RM) ^ 1.77
- (Bond Wi_RM) = 1.28 × (SPI/SGI, minutes) ^ 0.52
- (SPI/SGI, minutes) = 16667 × (A×b) ^ -1.39
- (A×b) = 844 × (SPI/SGI, minutes) ^ -0.66
Only SGI/SPI values below 250 minutes are used. Earlier published relationships were potentially biased by these unreasonably hard samples.
Preamble, data used
Today's instalment of "Alex is bored and is playing with data" looks at the relationship between the SAG tumbling test that goes by the name SPI (SAG Power Index, trademark of SGS) or the generic equivalent SGI (SAG Grindability Index). The result of this test is the time, in minutes, required to grind an ore charge to 80% passing 1.7 mm in a small batch (open-circuit) mill with a small ball charge.
The empirical equations are based on the Public Database of Grindability Testwork that was published last week on LinkedIn: https://www.dhirubhai.net/posts/alex-doll-66b57465_comminution-grinding-modelling-activity-6651521419916251136-_tuX
The tests that fall in the same size range as SPI & SGI are the Bond rod mill work index test and the two Drop Weight Tests (combining JK DWT and SMC Test?). The SAGDesign test would also fit this size range, but I haven't seen enough published values with both SAGDesign and the SPI/SGI metrics. See yesterday's post for explanation of why certain values are compatible: https://www.dhirubhai.net/posts/alex-doll-66b57465_grinding-modelling-comminution-activity-6653673682755559424-LyZ8
Graphs and regressions
First charts compare the SPI/SGI to the Bond rod mill work index. Non-standard rod mill results have been excluded from this analysis, only the wave-type rod mills are used.
SGI/SPI versus the A×b value from drop weights tests (both JK DWT and SMC Test) are best plotted using log-log axes.
Discussion - Equations not invertible
Two equations are obtained by plotting the overlapping sample data and using the regression tools built into the Libreoffice spreadsheet software.
People with sharp eyes will notice that the pair of equations (for a given pair of tests) are not invertible. What does this mean? Let's do some algebraic manipulation of the regression from the first figure:
- SGI = 0.89 × Wi ^ 1.77
- Wi ^ 1.77 = SGI ÷ 0.89
- Wi = (SGI / 0.89) ^ (1/1.77)
- Wi = 1.07 × SGI ^ 0.56
which does not match the equation predicted in the second figure:
- Wi = 1.28 × SGI ^ 0.52
Perhaps this lack of convertibility between the two equations is a result of the underlying statistics -- are any statistics fans able to comment on whether the minimizing of error terms goes wonky when dealing with logarithmic data?
How different are the predictions made by these different equations? They diverge by greater than five percent at the edges of the data (SGI > 300 or SGI < 25), but are within one percent in the middle, so pretty close for most ore types:
Conclusion
The SPI? and SGI metrics are empirically relatable to the Bond rod mill work index and A×b parameter of drop weight tests using the equations above. Be warned that these are not unique solutions, and there seems to be some statistical noise depending which way one organizes the regression.
The correlation coefficients (R2) are very good, 0.92 for the rod mill work index and 0.92 for the A×b if the SGI values above 250 minutes are excluded.
Use at your own risk -- same warning applies to all these regressions I've been posting.
Consultant at SAGMILLING.COM
4 年Further to discussions with Daniel Jordán (in PMs) and Paul Staples (below), the equation changes a bit if we filter out the really high SGI/SPI values -- here is the regression if we exclude values above 250 minutes. * (A×b) = 844 × (SGI, minutes) ^ -0.66
VP and Global Practice Lead, Minerals and Metals
4 年We have a much larger database, each ore has a different relationship and I would not use SPI above ~100. They measure different things. One is more indicative of impact breakage, the other attrition.
VP Operations at Base Metallurgical Laboratories Ltd.
4 年What is your opinion on the statistical relevance of supplementing an SPI dataset with RWI (waved liner) data?
Lab & Field Manager en Metso Perú SA
4 年Many thanks for sharing this data....if its possible i would like to Know the kind of deposit (skarn, porphyry? Etc) or if the samples come from the same domain. Additionally please confirm that the Whole data Was generated in the same lab....its complicated to get good relationships with data from several labs
Metallurgy Superintendent - Mineral Processing at Angloamerican Quellaveco
4 年Thanks Alex, The graph is interesting when the SPI is less than 100 minutes. But in your opinion, what is the reason for the high variability of the SPI when the Axb is less than 30. Have you found something interesting in this group of samples?, (Axb < 30)