Sampling Requirements for Auditing Diamond Recovery

Transcription

1 Sampling Requirements for Auditing Diamond Recovery Geoffrey J Lyman Materials Sampling & Consulting Australia Chris Prins De Beers Group Services, South Africa Godfrey Ngaisiue De Beers Marine Namibia Alex Gawanab De Beers Marine Namibia

2 Background De Beers Marine operates ships that mine offshore from the beaches of the Namibian coast The diamonds they recover are relatively large stones that have come down the Orange River The ships carry dense medium plants and x-ray diamond recovery circuits that treat material dredged from the ocean bottom They produced 600,000 carats in 2009 (120 kg ~ 3 x 20 litre bucket)

3 Objectives The objective was to determine the recoveries being achieved by the x-ray machines treating coarse, intermediate and fine diamonds This was to be achieved by installing additional machines to treat the tailings from the existing circuits The critical issue was the mass of tailings that would have to be treated in order to define the recoveries with a target accuracy

4 Problem Statement The size distribution of the stones to be recovered is provided as a mass distribution The number of stones recovered from a mining panel can be taken to be 10,000. We want to find the fraction of the tailings from the panel that will have to be retreated to define the recovery with a given accuracy

5 Solution Since the diamonds are all liberated, the problem can be solved by applying the properties of the Poisson distribution to the number of stones recovered in retreating the tailings The given mass distribution must be converted to a number distribution so we can deal purely in terms of numbers of stones

6 The Stone Distribution The distribution is plotted on probability paper If the line were straight, the distribution would be purely log-normal There is a gentle curve to the line, to which a 4 th order polynomial is fitted The cumulative distribution by mass can then be expressed as ln ln F g f g g sto n e m a ss n o rm a l d istrib u tio n fu n ctio n

7 Probability Density The Stone Distribution Making a transformation from stone mass to stone number, the density function is Mass Number g k ln ln 2 1 f g d f g exp 2 kg 2 d ln g n o rm a lisa tio n fa cto r Stone mass (carats) The plot shows the density functions by mass and by number as a function of stone mass

8 A Sample of Stones The average stone mass, by number, from numerical integration, is g m a x E g g g d g c t g m in For a sample of nominally N stones, the variance of the mass of stones in the sample is v ar 2 g g m a x m N g g dg m in N This expression was found using the fact that the variance of a Poisson random variable is equal to its expected value

9 The Recovery Estimate Because the recovery in the machines is high, the recovery estimate from reprocessing can be written as a random variable R 1 m a ss d ia m o n d s 'lo st' in ta ils m a ss d ia m o n d s re c o v e re d (e x c l ta ils) known exactly Now define the desired target SD for the mass of lost diamonds to be recovered from the tails to be a fraction f of the loss: SD R f 1 R

10 The Result The fraction of the tails to be reprocessed to keep the recovery estimate sufficiently accurate can now be expressed as m t R R f m 2 R m ass o f sto n es reco v ered fro m p an el (exc l tails) This result arises from the conversion of the distribution by mass to a distribution by number so that Poisson particle stats can be applied to estimate uncertainties The only additional assumption made was that recovery was independent of diamond size (which is a first approximation)

11 The Results 99.9% recovery will result in ~10 stones or ~7.59 cts in the tailings The distribution will be slightly skewed at this condition Full size distribution 10,000 stones per panel, m R =7490 cts

12 The Results 90% recovery will result in ~3 stones or ~21 cts in the tailings The distribution will be very skewed at this condition Coarse size distribution (2.84% by mass) 28 stones per panel, m R <212 cts

13 The Results 99% recovery will result in ~32 stones or ~46 cts in the tailings The distribution will be symmetrical at this condition Mids size distribution (61.7% by mass) 3231 stones per panel, m R <4621 cts

14 Conclusion The design of an auditing procedure for diamond x-ray recovery machines was required The application of Poisson sampling statistics and a bit of math provided a direct analytical solution to the problem in a practical form Thanks to De Beers Marine for the permission to publish this study