Grade efficiency curves reduced

The experimental results reported in this paper demonstrate the ability of a flat-bottom hydrocyclone to separate the coarse fraction of ammonium sulfate crystals from a slurry which contains crystals of a wide size range. It appears that the grade efficiency curve, which predicts the probability of a particle reporting to the underflow of the cyclone as a function of size, can be adjusted by a change in the underflow diameter of the hydrocyclone. These two observations lead to the suggestion to use hydrocyclone separation to reduce the crystal size distribution which is produced in crystallisers, whilst using a variable underflow diameter as an additional input for process control. 

For separators which operate with Rf > 0, the grade efficiency curve does not start at G = 0% (Figure 11JB). This is because the separator operation always gives a minimum efficiency almost equal to the flow ratio Rf. The reduced grade efficiency G may be obtained in the same way as the reduced total efficiency E ... 

Reduced grade efficiency curve and reduced cut size (c/50). 

If, in order to practically assess the performance of solid-fluid separators, a reduced grade efficiency curve is used, the particle size which gives 50% efficiency in such a curve is called the reduced cut size and is represented by x o-The maximum attainable efficiency related to particle size would be that minimum particle size with 100% probability of being reported to the underflow. Graphically, by extrapolating the end part of the curve to the horizontal axis, such size will be obtained. It has been proved that in practice, the maximum of the efficiency is around 98%, and the minimum size corresponding to this efficiency is represented by Xgg and known as the approximate limit of separation. 

A hydrocyclone, 44 mm in diameter, is used to separate a suspension of limestone in water. The density of the limestone is 2885 kg/m and the gravimetric separation efficiency was 28.34% while the partition coefficient was determined as 11.62%. Derive a grade efficiency curve to find out the reduced cut size, from the data given in Table 10.2. 

Following a procedure like the one described in Example 9.2, derive the reduced grade efficiency curve for the separation process. 

Note that the S-shaped grade efficiency curves do not necessarily start from the origin—in applications with a considerable underflow to throughput ratio (by volume) R, the grade efficiency curves tend to the value G x) = / f as X —> 0. This is a result of the splitting of the flow, or dead flux that carries even the finest solids into the underflow in proportion to the volumetric split of the feed. Section 3.4 discusses possible modifications to the efficiency definitions which account for the volumetric split and illustrate only the net separation effect. Such reduced efficiencies are widely used for hydrocyclones and nozzle-type disc centrifuges where large diluted underflows occur. 

This reduced efficiency concept is widely used in hydrocyclones the effect of this modification on the shape of the grade efficiency curve is shown in Figure 3.9 which uses the average curve of G x) from Table 3.2 (see section 3.3.1). It should be noted that the basic relationship between the total and grade efficiencies (equation 3.32) also holds for reduced efficiencies, so that ... 

This section describes a new and simple experimental method for obtaining the reduced cut size and the rest of the reduced grade efficiency curve of an operating separator. The method relies on feeding a known and fiiUy characterized slurry to the separator under test, and on measuring only two solids concentrations (in the feed and in the overflow), one static pressure differential (or some other flow rate-dependent variable) and the slurry temperature. These measurements are best done and logged by a personal computer, and, for hydrocyclones, have to be repeated at two different pressure settings. 

This forces the curve to pass through the origin as indicated by the second curve, G (x) in Figure 3.15. The reduced grade efficiency curve can, for some separators, be approximated by an analytical expression such as the one used in this method—see equation 3.51 in the following section. 

The reduced grade efficiency curves of separators can often be fitted by a cumulative log-normal function in the following form ... 

On the basis of the available experimental evidence it can be concluded that for a given cyclone design and low feed solids concentrations (say below 1% by volume) the shape of the reduced grade-efficiency curve is reasonably constant. At higher solids concentrations, it becomes dependent on the feed material and experimental measurement may be necessary. Alternatively, with smaller diameter cyclones, the log-normal law with a measured geometric standard deviation may also be used. Knowledge of the cut size is of course necessary in any of these cases to obtain the fiiU curve. 

Geometric standard deviation of the reduced grade efficiency curve of the cyclone Performance criteria Total coarse recovery by mass, r Total reduced mass recovery, r Actual cut size, xso Reduced cut size,... 

The last point needs explaining a little further. There are two reasons for deriving the sharpness index from the reduced grade efficiency curve rather than from the actual one. Firstly, the sharpness of cut is always to some degree affected by Rf and this way the effect is removed. Secondly, as can be seen later, the sharpness index for the actual grade efficiency curve sometimes cannot have a value because the whole of the curve may be greater than 25% and X25 cannot be determined. 

The last conclusion is qualitatively the same as that for the overflow series in that they both sharpen the cut. Even quantitatively, when using a symmetrical sharpness index derived from the reduced grade efficiency curves, one obtains very much the same results for the underflow series as contained in... 

Figure 16.12 The reduced grade efficiency curves of single-stage, two-stage and three-stage series on underflow plotted against dimensionless particle size...

The size corresponding to 50% on the reduced grade efficiency curve G (x) is referred to as the "reduced cut size" - see Fig.l. Most mathematical descriptions of the performance of hydrocyclones or sedimenting centrifuges are in terms of the reduced cut size. 

Equation (17), obtained from Equation (10), shows that it is possible to estimate the total efficiency of cell separation when the curves of grade efficiency and feed size distribution are known. Based on Equations (16) and (17), it is possible to obtain a similar result for the reduced total efficiency (Equation 18). 

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