View source for Template:Model theory (Text, Gravity Concentrator, Middlings)

=== Middlings ===

Gravity concentrators such as jigs, spirals and shaking tables produce a bed or band of partially stratified components at the point of discharge. Portions of the bed or band are then typically directed to product streams by a physical device, such as a weir, 'splitter' or 'cutter'. These devices are usually adjustable, and can be arranged to recover arbitrary fractions of the bed or band.

From a physical standpoint, adjusting the discharge device to recover more of the bed or band has the effect of recovering the both the portion from the previous position ''plus'' the portion in between the previous and new positions. As more mass is recovered by this process, the partition curve effectively 'shifts upwards'. The partition curve is thus representing the ''cumulative recovery'' of mass from all positions between the beginning of the bed/band and the discharge device position.

Mathematically, the partition curve generated by such a gravity concentration method should also be considered a cumulative recovery of mass to concentrate. When multiple product streams exist, e.g. concentrate and middlings, the partition of components to each individual product stream will be the difference between the cumulative partition curves at each product stream position.{{King et al. (1992)}} That is,

:<math>
y_{ij,s} = 
\begin{cases}
Y_{ij,s} & s=1\\
\max \left ( 0, Y_{ij,s} - Y_{ij,s-1} \right ) & s>1\\
\end{cases}
</math>

where:
* <math>s</math> is the index of the product stream, i.e. <math>s=1</math> is the first concentrate stream, <math>s>1</math> are subsequent lower-grade concentrate or middlings streams
* <math>y_{ij,s}</math> is the mass fraction of particles in the feed stream in size class <math>i</math> and density class <math>j</math> which are partitioned to the product stream <math>s</math> (frac)
* <math>Y_{ij,s}</math> is the cumulative mass fraction of particles in the feed stream in size class <math>i</math> and density class <math>j</math> which are partitioned to all the products streams up to and including <math>s</math> (frac)
* The <math>\max</math> term prevents negative partition values

The cumulative recovery formulation described above is physically and mathematically distinct from ''staged'' recovery processes which apply partition curves to the unrecovered streams of previous partition steps.