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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{{\rm oc}s} = \begin{cases} \exp (- n_L L f_{\rm o} \dfrac {\big [D_1 - s \big ] \big [D_2 - s \big ]}{D_1 D_2}) & s < D_2\\ 1.0 & s \geq D_2\\ \end{cases} }

where:

• $s$ is the diameter of a particle (mm)
• $E_{{\rm {oc}}s}$ is the corrected fraction of particles of size $s$ in the feed reporting to the oversize stream (frac)
• $n_{\rm {L}}$ 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:

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

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 ${\bar {d}}_{\rm {AR}}=1$.

An average value of the partition is required when using size intervals. Whiten (1972) suggests the following averaging approach:

$E_{{\rm {oc}}i}={\frac {1}{(s_{i-1}-s_{i})}}\int _{s_{i}}^{s_{i-1}}E_{{\rm {oc}}s}\mathrm {d} s$

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)
• $s_{i}$ is the size of the square mesh at size interval $i$ (mm) that particles are retained on, in descending order such that $s_{i}<s_{i-1}$

Numerical integration is applied to determine the average partition values of each size interval. Whiten's (1972) method is particularly useful when the partition curve is very steep, as may be the case for screens operating at high efficiency.

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)
• ${\bar {d}}_{i}$ is the geometric mean size of the size interval $i$ (mm)
• $\lambda$ is the size constant