Grade efficiency curve equations

Cyclone Efficiency. Most cyclone manufacturers provide grade-efficiency curves to predict overall collection efficiency of a dust stream in a particular cyclone. Many investigators have attempted to develop a generalized grade-efficiency curve for cyclones, eg, see (159). One problem is that a cyclone s efficiency is affected by its geometric design. Equation 15 was proposed to calculate the smallest particle size collectable in a cyclone with 100% efficiency (157). 

This equation is for Eigure 9 cyclone dimension ratios. The term the effective number of spirals the gas makes in the cyclone, was found to be approximately 5 for Lapple s system (134). The soHd line grade-efficiency curve of Eigure 7 is also used with Lapple s cyclone, which is a somewhat taller, less compact cyclone than many commercial designs. 

The definition of the grade efficiency (Equation 9) is similar to the total efficiency definition, but it applies only to a given bioparticle size. Figure 11.5 shows a typical grade efficiency curve ... 

Through Equations (10), (13), and (15), it is possible to obtain the grade efficiency curve based on the total efficiency E and the size distributions of two out of three streams (j,yp andyc)-... 

These equations are used to determine the grade efficiency of a classifier provided the total efficiency and the size distributions of two of the streams are known. Results are usually plotted as grade efficiency curves of G (jc) or Gj(x) against jc [3]. Since the classifier separates on the basis of Stokes diameter it is preferable to carry out the size determinations, for grade efficiency evaluations, on the same basis. 

If the gas stream entering the unit consists of a distribution of particles of various sizes, then frequently a fractional or grade efficiency curve is specified for the settler. This is simply a curve describing the collection efficiency for particles of various sizes. The dependency of E on arises because of the n, term in the above equations. 

If the grade efficiency curve is now to be plotted against particle size for a material with a density difference of (ps — Pi) = 1000kgm , equation 3.12... 

A simple way has recently been suggested of determining jcso without the need for the whole grade efficiency curve. If G(x) in equation 3.17 is equal to 0.5, it can be shown that the cut size X50 corresponds to the point on a plot of (F — 2EjFc) at which ... 

Equations 3.22, 3.23 and 3.24 require plotting the composite functions of two solid size distributions and the total efficiency and finding the maximum (see the worked example elsewhere ), which requires no differentiation and can be performed easily in industrial production situations. Naturally, the test information necessary for this calculation is the same as for the grade efficiency curve itself two particle size distributions and the total efficiency Ej. 

If the grade efficiency curve can be regarded as a characteristic parameter of a separator for particular conditions (flow rate, viscosity of liquid etc.), this curve can be used to determine the total efficiency that can be expected to be obtained with a particular feed material under the same conditions. The size distribution dF(x)/dx of the feed material must be known, then, from equation 3.17 ... 

If both the feed size distribution F(x) and the grade efficiency curve G(x) can be approximated by an analytical function, the integration in equation 3.32 can be done analytically. Equation 3.25 can, for example, be used for log-normal functions—see section 3.3.1.5. 

The grade efficiency curve G(x), which in its unmodified form (defined by equation 3.15) also obscures the existence of a volumetric split of the flow, can be modified in the same way ... 

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 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. 

Equation 7.9 gives the typical S-shaped grade efficiency curve (see Figure 7.3) the limit of separation that appears in equation 7.9b may be determined from equation 7.6. 

An important parameter that can be derived from equation 7.7 (or 7.9) is the size corresponding to 50% on the grade efficiency curve, i.e. the equiprob-able size or cut size X50 (see chapter 3, Efficiency of Separation ). The corresponding radius r o is the one that splits the annulus between r and r into equal areas hence... 

As can be seen from the graph, Bradley s model gives conservative predictions of efficiency, mainly because he applied the approximation in equation 7.20, which was originally fitted to data obtained at radii between 1 and 2 cm, to radii up to rs = 2.223 cm where large discrepancies occur this leads to underestimates of separation efficiency. Bradley s model is still useful because it gives a lower estimate of efficiency, the actual grade efficiency curves are usually found to lie between those predicted by equations 7.9 and 7.21. 

Equations 7.32 and 7.33 fully describe the theoretical grade efficiency curve which, as can be seen, is independent of the spacing between the discs but depends on the number of discs n. Equation 7.32 is a simple parabola and the cut size X50 can be determined from equation 7.32 since G(x5o) = 0.5 hence... 

Equations 15.1 and 15.2 can be used to compute the grade efficiency curves for each system in Figures 15.1 and 15.7. This calculation is best performed in a spreadsheet where each line in Table 15.1 is processed by the two equations, yielding two additional columns. These are given in Table 15.2 where the given values of particle size x and differential size distribution in the feedy(x) are repeated in the first two columns. 

If the grade efficiency curves are known for all stages in a system, an equation for the grade efficiency of the whole system may be derived from mass balance around the arrangement. Reference 1 gave, for the first time. 

Figure 9.6 Grade efficiency curve described by Equation (9.22) for a cut size X50 = 5 im...

The grade efficiency curve can be determined by measuring the total efficiency and the particle size distribution of any two of the three streams (feed, underflow, and overflow). One of the following equations can be used for calculating the grade efficiency ... 

In either of the alternatives, no dilution is necessary and, furthermore, the separator itself can be tested using the same instrumentation as installed for the monitoring of particle size. This in effect relies on the same basic equation (eqn.lO) but this time, and 0,3 are known for the test suspension fed to the separator and x so (the cut size) and (the standard deviation of the grade efficiency curve) are determined from the tests. 

Equation (5.2.2) represents one functional form for representing the grade-efficiency curve (GEC). The writers own analysis on numerous commercial and laboratory cyclones reveals that the form of Eq. (5.2.2) describes some cyclone geometries quite well, especially smooth, well-designed laboratory cyclones. The exponent 6.4 is, however, a little larger than the values t3q>i-cally observed in some large-scale, refractory-lined, commercial cyclones and in some poorly designed small-scale cyclones. In these latter cases, the exponent typically lies between 2 and 4. 

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