Converting Between Overall Efficiency and Cut-size

We often have occasion to convert between the overall efficiency 77 and the cut size 0 50, for instance, when we wish to compare an efficiency determined by testing with the cut-size predicted by a model. There are two ways of doing this a simple approximation and the precise way. 

The simple approximation is to assume the cyclone to have a sharp cut at X50, i.e. that all material below X50 is lost and all material above is collected. If the cumulative undersize distribution of the feed is Ff x), then (see Fig. 3.2.2)  

The precise way is to use the entire grade-efficiency curve for the cyclone, r] x). The fraction of feed lying within the band x — j2 dx and x - - 1/2 dx, is captured with a fractional efficiency of r (x)ff x)dx. The total fraction of the feed captured is therefore  

Experience shows that the approximate method (Eq. 3.2.7) comes out surprisingly accurate, even when the cut of the cyclone is far from sharp. This has very practical implications and is often used when one just needs a rough estimate of overall separation efficiency. 

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