Description
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Model theory
Whiten-Beta
The Whiten-Beta expression for partition to overflow is:[1]
![{\displaystyle E_{\rm {oa}}({\bar {d}}_{i})=C\left[{\dfrac {\left(1+\beta \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)(\exp(\alpha )-1)}{\exp \left(\alpha \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)+\exp(\alpha )-2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/692eadc8e565caa11dfb67b8220c1b59aeb45652)
where:
is the index of the size interval,
,
is the number of size intervals
is the fraction of particles of size interval
in the feed reporting to the overflow stream (frac)
is the geometric mean size of particles in size interval
(mm)
is the corrected size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
is the fraction of feed liquids (or fines) split to overflow (frac)
is a parameter representing the sharpness of separation
is a term introduced to accommodate the so-called fish-hook effect, and controls the initial rise in the efficiency curve at finer sizes
is computed to ensure the Whiten-Beta function preserves the definition of
in the presence of the fish-hook, i.e.
at 
The value of
is not an input parameter and is computed numerically, by iteration, since by rearrangement:
![{\displaystyle \beta ^{*}={\dfrac {\ln {\big [}\exp(\alpha )+2\beta \beta ^{*}(\exp(\alpha )-1){\big ]}}{\alpha }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/373fc264d9131d512a86789d6b285ee857d80e40)
When the value of
is zero, the Whiten-Beta equation reverts to:
![{\displaystyle E_{{\rm {oa}}i}=C\left[{\dfrac {\exp(\alpha )-1}{\exp \left(\alpha {\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)+\exp(\alpha )-2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f6ed566ef901b01a52c36cc0aa24e41cca3206)
Excel
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SysCAD
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References
- ↑ Napier-Munn, T.J., Morrell, S., Morrison, R.D. and Kojovic, T., 1996. Mineral comminution circuits: their operation and optimisation. Julius Kruttschnitt Mineral Research Centre, Indooroopilly, QLD.