Technical Separation Efficiency: Definition and Derivation

Based on Jowett (1975) and Jowett and Sutherland (1985), as the basis for the definition of separation efficiency we take the following:

$S E = \frac{Amount perfectly separated}{Amount theoretically perfectly separable}$ $S E = \frac{Amount perfectly separated}{Amount theoretically perfectly separable}$

A solution is derived from the cumulative recovery–cumulative total solids (or weight) recovery (R–W) method of presenting separation data. In this method of data presentation, the slope of any line represents a grade, or more precisely grade divided by the feed grade. This can be shown by noting:

$Slope = \frac{Δ R}{Δ W} = \frac{Δ (C c)}{F f} / \frac{Δ C}{F} = \frac{c}{f}$ $Slope = \frac{Δ R}{Δ W} = \frac{Δ (C c)}{F f} / \frac{Δ C}{F} = \frac{c}{f}$

This property is exploited in the derivation of SE. (As a consequence of this property, the plot is sometimes referred to as a grade-gradient plot.)

Figure A.III.1 shows a representative R–W operating line, OMQ, with the necessary constructions: line OQ is the “no separation” line (c=f); OPQ represents perfect separation; and lines J(I)M and KP are parallel to the no separation line.

Figure A.III.1 Generic recovery–weight recovery (R–W) plot with constructions to determine separation efficiency.

We can now define some features of the plot.

Weight recovery (or yield) at P, W_P:

By inspection, when metal recovery is 1 (100%) at pure mineral grade then the weight recovered, W_P, must be the weight of mineral contained in the feed, that is, the feed mineral grade. Using the same symbols as in Chapter 1, then if f is the feed metal grade and m is the content of metal in the mineral then the feed mineral grade is f/m.

Thus

$W_{P} = \frac{f}{m}$ $W_{P} = \frac{f}{m}$

Operating point M:

From the construction, we can consider point M to be made of part perfect separation, part feed by-passed to the concentrate, that is:

$O M = O I + I M$ $O M = O I + I M$

where OI is the perfectly separated part, and IM is the by-passed feed part.

Separation efficiency:

Based on the definition above

$S E = \frac{O I}{O P}$ $S E = \frac{O I}{O P}$

Derivation in terms of R, etc.

By similar triangles we can write

$S E = \frac{O I}{O P} = \frac{O J}{O K} = \frac{J}{K}$ $S E = \frac{O I}{O P} = \frac{O J}{O K} = \frac{J}{K}$

J and K:

$\begin{matrix} slope J M = 1 = \frac{R_{M} - J}{W_{M}} \\ thus J = R_{M} - W_{M} \\ slope K P = 1 = \frac{1 - K}{W_{P}} \\ thus K = 1 - W_{P} \end{matrix}$ $\begin{matrix} slope J M = 1 = \frac{R_{M} - J}{W_{M}} \\ thus J = R_{M} - W_{M} \\ slope K P = 1 = \frac{1 - K}{W_{P}} \\ thus K = 1 - W_{P} \end{matrix}$

Consequently:

$S E = \frac{R_{M} - W_{M}}{1 - W_{P}} = \frac{R_{M} - W_{M}}{1 - f / m}$ $S E = \frac{R_{M} - W_{M}}{1 - W_{P}} = \frac{R_{M} - W_{M}}{1 - f / m}$

General case:

$S E = \frac{R - W}{1 - f / m}$ $S E = \frac{R - W}{1 - f / m}$

Substituting for R=Cc/Ff and W=C/F

$\begin{array}{l} S E & = \frac{C}{F} \frac{m}{f} \frac{(c - f)}{(m - f)} = \frac{C}{F} (\frac{c}{f} - \frac{m - c}{m - f}) \\ = \frac{C}{F} \frac{m}{f} \frac{(c - f)}{(m - f)} \end{array}$ $\begin{array}{l} S E & = \frac{C}{F} \frac{m}{f} \frac{(c - f)}{(m - f)} = \frac{C}{F} (\frac{c}{f} - \frac{m - c}{m - f}) \\ = \frac{C}{F} \frac{m}{f} \frac{(c - f)}{(m - f)} \end{array}$

The result, starting from the above definition of separation efficiency, is thus equivalent to Eq. (1.5), which in turn means SE is equal to difference in recoveries (Eq. (1.2)).

Maximum separation efficiency:

The maximum can be found by trial and error from the available recovery data for the two components, as illustrated in Chapter 12, Figure 12.58 (see also Appendix VI). It can also be found by inspection of Figure A.III.1, noting that the maximum SE will correspond to when OJ is maximum. This will occur when the line constructed through J parallel to the no separation line is tangent to the operating line; in other words, the maximum SE corresponds to the point on the operating line where the increment in grade (the tangent) is equal to the feed grade. This result has led to attempts to include the increment of grade concept in flowsheet design to optimize separation (see Chapters 11 and 12).

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Table of Contents for Appendix III. Technical Separation Efficiency: Definition and Derivation

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Technical Separation Efficiency: Definition and Derivation

Table of Contents for
Appendix III. Technical Separation Efficiency: Definition and Derivation