Part of Gy’s work developing his theory of sampling was a formula for estimating the variance of the fundamental sampling error from the characteristics of the lot and sample. As variance is the square of standard deviation we can then estimate the fundamental sampling error at a 95% or a 99% confidence level at any stage in the sample preparation process. This formula can be used to help design optimal sampling systems and protocols. I have used it to help formulate the financial justifications for changes to old incorrect sampling system.
This is Gy’s Formula
σ2FSE = variance of the fundamental sampling error
C = a modifying factor that describes the material being sampled
d= top particle size in cm
Ms= sample mass (g)
Ml= mass of the lot (g)
Working out C
Imagine a pile of rocks like this in a variety of sizes, like a stockpile. What observable characteristics can you think of that might help you describe it to someone else?
Think of these characteristics. They are the same ones you would need to know to work out the variance of the fundamental sampling error.
- fragmentation or size range of the fragments
- fragment shape
- grade (true value)
- size of ore grains (liberation size)
- proportion of ore to waste
- ore mineral
- nature of the waste material
- size of the stock pile
These are the factors that are taken into account when determining C
C = fgml
so Gy’s formula can be written as
f= particle shape factor (describes the shape of the particles)
g= granulometric factor (describes how much variation there is in the size of particles)
l = liberation factor (how close to liberation the material has been ground)
m = mineralogical composition factor (describes how much of a rock is made up of the element of interest at a given grade)
So how do you work out all these modifying factors Gy came up with?
Fortunately there is a lot of experimental data for f and g which can be used for a quick estimate.
Working out f
Ranges from 0-1 in most circumstances and can be thought of as describing how cubic a particle is.
A quick look with a microscope is generally all that is needed to determine the shape factor. This can usually be done by reference to results from experiments. The following are some general rules.
A true sphere is 0.523
Ores are around 0.5 e.g. pyrite depending on the form it takes is 0.495-0.514
Needle shaped minerals are 1-10
Flaky minerals like biotite are around 0.1
Soft materials are around 0.2
Calibrated samples of the material where the size is constant can be used to experimentally determine particle shape factor if no prior work exists.
Working out g
Since this describes how varied the size of the fragments in a lot are it’s a lot like describing the fragmentation of material after a blast.
It ranges from 0-1
everything is the same size=1
no fragments the same size=0
Ores are generally 0.25
Calibrated material (stuff that has been screened using a sieve )=0.5
Naturally calibrated materials like grains=0.75
Working out l
Describes how much of the ore mineral we could expect to be completely separated from the waste.
We can work this out using the particle top size and the liberation size.
Liberation size is the size where the ore minerals are freed from the waste.
Liberation size can be determined experimentally using things like scanning electron microscopes, but it’s often easier on a mine site to ask the mill metallurgists for the figure they use.
The particle top size can be determined using the size of the holes in the mesh that 95% of the material passes through or various other size testing methods.
L = liberation size (cm)
d= particle top size
Working out m
Describes how much of a rock is made up of the element of interest at a given grade and corrects for the fact that density is not evenly distributed between ore and waste material.
a = proportion of element at a set grade as a decimal
r = density of ore mineral
t= waste/gangue density
a is the decimal proportion of mineral required to give the grade.
Mineral data for finding a can be found at the following website
After you work out all of that, for most of the sampling stages the only thing that will change are the top particle size d, and the masses of the lot and sample.
I’ll put an example of this in the next post.
If you take the square root of Gy’s formula, you have the standard deviation of the fundamental sampling error. With this you can work out what the grade variation is going to be at the grade used to determine m. This can be done to 2 or 3 FSE standard deviations by multiplying the grade by the standard deviation you want to use. These will also give you an estimate of the 95% or 99% confidence levels if that’s what your boss wants to know about.