The sorting of waste for a circular economy : sampling when (very) few particles have (very) high concentrations of contaminant or valuable element

The measurement of elements in numerous individual particles (≥ 1 cm²) by a portable X-ray fluorimeter is used to review a part of the sampling theory of granular solid waste for environmental studies and the circular economy. The paper addresses the case when the concentration of element or substance is not related to the grain size. The key concept (from the binomial law) is the number of particles that must be present in a portion of matter to be representative of a larger portion of matter. This number depends on the frequency of particles having the studied property, and on the desired variability of this property. In all cases, the lowest achievable variability is the analytical variability with the smallest possible test portion (results cannot be less variable). The studied property can either be the presence of an element or a substance, or the presence of an element or a substance at a given concentration. Those concepts are the basis of the existing sampling standards but are not presented as such. As a result, the equations of these standards are not easy to understand and, to our knowledge, rarely used to calculate the mass of a representative sample. When the distributions of concentrations are skewed by (very) large values, the last centiles of concentration tremendously increase the observed mean concentration of a waste heap or flow, and a representative sample must include these last centiles of particles for a proper characterizing and sorting of waste and secondary raw material for the circular economy. Data of centiles of concentrations per particle and laboratory analytical variability are presented. The resulting recommended number of particles that should be present in a sample at any scale from the waste stream (thousands of tons) to the test portion (frequently less than one gram) is estimated at 100 000. Some published sampling plans (from the waste stream to the laboratory sample) and analytical standards (from the laboratory sample to the test portion) are then reviewed for the number of particles. It is crucial to measure the mean mass of the particles to sample, from the granulometric distribution of the particles, and the bulk density, to determine the weight and volume of 100 000 particles. If there is a fine fraction (< 63 μm or even < 1 mm), the recommended mass or volume complies with the requirement of n ≥ 100 000. When there is no fine fraction, like for some WEEE plastic scraps, the volume recommended in technical specification and standard for laboratory sample can’t have enough particles for p = 0.001, but well for p = 0.1 for plastics from small household appliances or higher p for plastics from fluorescent lamps. These p values must be verified for these plastics but are probably not unrealistic for the unsorted fraction. On the other hand, using the equation of the sampling standards overestimates the mean mass of particle when fines are present. Another application is the evaluation of the number of particles to measure individually, in order to calculate the fraction of particles trespassing a given concentration of an element, and the confidence interval of that fraction