Partitions

Description

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Model theory

Whiten-Beta

The Whiten-Beta expression for partition to overflow is:

[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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