Partitions
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Description
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Model theory
Whiten-Beta
The Whiten-Beta expression for partition to overflow is:[1]
- [math]\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 ] }[/math]
where:
- [math]\displaystyle{ i }[/math] is the index of the size interval, [math]\displaystyle{ i = \{1,2,\dots,n\} }[/math], [math]\displaystyle{ n }[/math] is the number of size intervals
- [math]\displaystyle{ E_{\rm oa}(\bar d_i) }[/math] is the fraction of particles of size interval [math]\displaystyle{ i }[/math] in the feed reporting to the overflow stream (frac)
- [math]\displaystyle{ \bar d_{i} }[/math] is the geometric mean size of particles in size interval [math]\displaystyle{ i }[/math] (mm)
- [math]\displaystyle{ d_{\rm 50c} }[/math] is the corrected size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
- [math]\displaystyle{ C }[/math] is the fraction of feed liquids (or fines) split to overflow (frac)
- [math]\displaystyle{ \alpha }[/math] is a parameter representing the sharpness of separation
- [math]\displaystyle{ \beta }[/math] is a term introduced to accommodate the so-called fish-hook effect, and controls the initial rise in the efficiency curve at finer sizes
- [math]\displaystyle{ \beta^* }[/math] is computed to ensure the Whiten-Beta function preserves the definition of [math]\displaystyle{ d_{\rm 50c} }[/math] in the presence of the fish-hook, i.e. [math]\displaystyle{ E = 0.5 C }[/math] at [math]\displaystyle{ d_{\rm 50c} }[/math]
The value of [math]\displaystyle{ \beta^* }[/math] is not an input parameter and is computed numerically, by iteration, since by rearrangement:
- [math]\displaystyle{ \beta^* = \dfrac{\ln \big[ \exp (\alpha) + 2 \beta \beta^* (\exp (\alpha) - 1) \big]}{\alpha} }[/math]
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.