Grade efficiency models

Practically all cyclone performance data have been related to a present cyclone set of geometric ratios. One model for cyclone grade-efficiency curves has been tested against reported commercial cyclone efficiencies (159). A good fit was obtained. 

A simple, plug-flow model of the separation in a settling tank without flocculation gives the grade efficiency in the following form... 

Unlike the theories reviewed in the previous sections, the model by Bloor et al. does not lead to any simple correlations for the cut size or the grade efficiency curve, but it is the first time that any direct proof of the crowding theory is given. On the basis of an actual set of conditions studied, the authors give a quantitative example if a cyclone operates satisfactorily at 5% solids concentration and this is increased to 15%, the underflow rate must be increased by a factor of 1.6 in order to prevent overcrowding and possible blocking of the underflow orifice. 

The comparison of the Rajamani model with experiments is very good, done via grade efficiency curves in refs. 39 and 42. One paradox in this is that the grade efficiency concept is only useful if grade efficiency is independent of the particle size distribution in the feed. If, as seems to be true at high concentrations, this independence is no longer valid and we have to resort to... 

The method of grade efficiency derivation will be demonstrated using a much simpler model in which the velocity profile in the liquid shell is assumed to be uniform, as in the so-called plug flow ... 

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. 

One appropiate way to express the efficiency of a cyclone is in terms of a grade efficiency curve. In this case the efficiency is calculated for every particle size class and the resulting curve is an S-type curve with efficiency values between 0 and 1. In the test rig under investigation we can apply three models of grade efficiency calculation methods ... 

The errors for models 1 and 2 are clearly the highest. Model 3 has a smaller error. Model 3 offers the best potential for the calculation of the grade efficiency curve. It is less sensitive to systematic errors since they act in the same direction. 

The results of the grade efficiency calculations using equation (3), model 3, are shown in Figure 6. 

As the best model for grade efficiency calculations we applied model 3 in which the catch and probe 2 sample should be known. This model resembles the procedure proposed by Hermann and Leschonski [3], as a correction of the mass balance is applied in this calculation model as well. It appears from our experience that the size analysis of the probe 2 sample must be carried out very accurately. Enough sample should be collected on the filter in order to perform reliable Coulter Counter analysis. 

The models agree well in terms of critical particle diameter or cut size. The grade-efficiency curve of Mothes-Loffier is obviously much flatter than that of Barth. We refer to our discussion above, where we state that Barth s index of 6.4 in Eq. (5.2.2) is in the high end of the range, and best suited for smooth, well-designed cyclones. In our experience the index for many older units of poorer design often lies between 2 and 4, which gives a slope much more in line with the model of Mothes and Lofiier. 

Barth Efficiency model valid for all cyclones and swirl tubes (but see Table 4.5.1). Equilibrium-orbit model. Calculates the cut size, then fits an empirical grade-efficiency curve through it. Time-of-flight model. Derivation considers particle motion, but the final model relates cut size to pressure drop. 

Fig. 5. A.2. Predicted grade-efficiency curves for the cyclone in Fig. 4.A.I. The cut size predicted by the Rietema model is also shown...

Figure 6.A.3 presents the model s predictions at 4.5 and 31.7 g/m solids loadings corresponding to two of the four sets of experimental grade-efficiency data. Figure 6.A.4 presents the Muschelknautz predictions of the particle size distribution of the overhead dust fraction at two different solids loadings.

Fig. 6. A.3. The Muschelknautz model predictions of grade-efficiency at two different solids loadings, compared with experiment. Inlet velocity 10 m/s...

The model predictions of the cyclone s grade-efficiency curve are shown in Fig. 6.A.6. The model is seen to predict actual performance quite well for particle sizes greater than about 2 pm but overpredicts actual performance for smaller sized particles. In this case, the model predicts a mass loading effect for the smaller sized particles that is larger than what was observed. 

The model s prediction of the classification portion of the cyclone s grade-efficiency curve (i.e., that excluding the solids loading effect) is shown in Fig. 6.A.8. Here, efficiencies are plotted as a function of the dimensionless particle ratio x/x o, where x is the particle diameter and X50 the cyclone s computed cut size. The model is seen to predict measurements reasonably well although the slope of the predicted s-shaped grade-efficiency curve (m = 5) is greater than that of the experimental data. 

Fig. 7.3.3. Grade-efficiency curves from Derksen (2003) at different points in time. The predictions for Stk o of the models of Barth and Mothes and Loffler (Chap. 5) are also indicated...

A model cyclone of diameter D = 0.2 m operating at an inlet velocity = 15 m/s at ambient conditions on a chalk powder of density 2700 kg/m , at low solids loading, is by testing found to have the grade-efficiency curve shown in Fig. 8.B.I. The cyclone pressure drop was found to be 950 Pa. 

We note from the experimental data reported above that the model s cut-point diameter, X50, is 0.98 yum. We do not have data at the same Re for model and prototype, so we will make use of the approximations mentioned in the main text. Rein is large enough in both model and prototype to assume that the grade-efficiency is about the same in the two cyclones for the same value of Stk, and that their Eu values are the same. 

We begin by scaling the entire grade-efficiency curve. We calculate Stk corresponding to the particle sizes in the model data. These values of Stk... 

Fig. 8.B.I. Experimental grade-efficiency data for a laboratory model cyclone Table 8.B.I. Relevant physical and operational data for model and prototype...

We see from the figure that we can expect all catalyst particles greater than about 10 pva to be completely captured in the industrial unit. We also see that the grade-efficiency curve for the prototype has the same s-shape form as that of the model on the logarithmic scale. 

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