Template:Model theory (Text, Whiten and White Partition Curve)

$E_{{\rm {oc}}i}={\begin{cases}\exp(-n_{L}Lf_{\rm {o}}{\dfrac {{\big [}D_{1}-{\bar {d}}_{i}{\big ]}{\big [}D_{2}-{\bar {d}}_{i}{\big ]}}{D_{1}D_{2}}})&{\bar {d}}_{i}<D_{2}\\1.0&{\bar {d}}_{i}\geq D_{2}\\\end{cases}}$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $E_{{\rm {oc}}i}$ is the corrected fraction of particles of size interval $i$ in the feed reporting to the oversize stream (frac)
• ${\bar {d}}_{i}$ is the geometric mean size of the size interval $i$ (mm)
• $n_{\rm {L}}$ is the number of trials per unit length parameter (/m)
• $L$ is the length of the screening area (or panel) in the direction of flow (m)
• $f_{\rm {o}}$ is the fraction open area of the screen deck/panels (m²/m²)
• $D_{1}$ is length (the longer side) of the screen deck/panel aperture (m)
• $D_{2}$ is width (the shorter side) of the screen deck/panel aperture (m)

Whiten and White's equation was modified by Dehghani et al. (2002) to include a term for irregularly shaped particles, and is generalised to:

$E_{{\rm {oc}}i}={\begin{cases}\exp(-n_{L}Lf_{\rm {o}}{\dfrac {{\big [}D_{1}-{\sqrt {2}}{\bar {d}}_{i}\cos \theta {\big ]}{\big [}D_{2}-{\sqrt {2}}{\bar {d}}_{i}\sin \theta {\big ]}}{D_{1}D_{2}}})&{\sqrt {2}}{\bar {d}}_{i}\sin \theta <D_{2}\\1.0&{\sqrt {2}}{\bar {d}}_{i}\sin \theta \geq D_{2}\\\end{cases}}$

where:

$\theta =\arctan d_{\rm {AR}}$

and $d_{\rm {AR}}$ is the representative aspect ratio of the particle population, the ratio of the second longest to the longest dimensions of a particle (mm/mm)

The aspect ratio property, $d_{\rm {AR}}$, allows for the balanced screening of 'flaky' or 'elongated' particles on slotted meshes. Dehghani et al.'s relation reduces to Whiten and White's original equation when ${\bar {d}}_{\rm {AR}}=1$.

Firth and Hart (2008) suggested a simple modification to a partition curve to account for the observed entrainment of fine particles in an oversize stream, with decreasing probability as particle size increases:

$E_{{\rm {oa}}i}=E_{{\rm {oc}}i}+\left(1-E_{{\rm {oc}}i}\right)\cdot R_{\rm {f}}\exp(-\lambda {\bar {d}}_{i})$

where:

• $E_{{\rm {oa}}i}$ is the actual fraction of particles of size interval $i$ in the feed reporting to the oversize stream (frac)
• $R_{\rm {f}}$ is the fraction of the finest particles or feed liquids split to the oversize stream (frac)
• $\lambda$ is the size constant