O dx dx Steady state continuous particle separator underflow (coarse) Mc dFc(x), U dx Fig. 2 Schematic diagram of single stage continuous particle separator/classifier. A total or overall efficiency ET of the process is defined as the ratio of mass Mc of all the particles recovered in the coarse product to the mass M of solids in the feed. ET = Mc /M [12a] or ET = 1 ? Mf /M [12b] dF ET dFc (1? ET )dFf [12c] ? = ? + ? dx dx dx.
This holds for fractions of particular size and cumulative percentages corresponding to any given size in the following (integrated) forms of this expression:- F(x2) ? F(x1) = ET((Fc(x2) ? Fc(x1)) + (1? ET )((Ff (x2) ? Ff(x1)) or F(x) = ET Fc(x) + (1? ET)Ff (x) [12d] Thus if ET is known then one of the three size distributions can be calculated from the other two.
More usually ET can be obtained from the three size distributions. This latter approach also facilitates an analysis of errors in sampling and analysis.
Rearrangement of the last equation gives. F(x) ? Ff (x) ET = ????? [12e] Fc(x) ? Ff (x) A plot of F(x) ? Ff(x) as a f[Fc(x) ?
Ff(x)] for various sizes (x) should produce a straight line of gradient ET. Random scatter of data provides an indication of errors. Non linearity indicates possible comminution or agglomeration of the feed material. 4. Grade Efficiency Definition G(x) = Mc(x)/M(x) for all x in the appropriate range. It therefore is a f(x), whereas ET is defined for the total mass.
The performance of most equipment is dependent on the particle size of the feed. (Different sizes are separated with different efficiencies. ) The overall efficiency therefore may serve only limited purposes (i. e. problems related to material used in the test for ET).
However, a curve can be produced which provides the mass efficiency for every particle size. This is known as the gravimetric grade efficiency curve (G(x) vs x, Fig. 4). This is usually determined by experiment but can be predicted for some separators. This is often independent of the solid size distribution and density and is constant for a given set of operating conditions (? , Q, M). It is important that the characteristic particle (size) is appropriate to that of the separation method employed. e. g. continuous separation in rotary screens – sieve analysis settling in gravitational/centrifugal fields – sedigraph/sedimentation analysis.
Translation of G(x) curves to those for solids and/or media of other properties will subsume all previous assumptions and may therefore not possess accuracy. Example: In the case of particle sedimentation, conversion of a G(x) curve for solids of density ? 1 suspended in a liquid of viscosity ? 1, to a system of small particles of density ? 2, suspended in a liquid of viscosity ? 2 can be made using Stokes’ law from which:-  This subsumes all the assumptions of Stokes’ law and assumes that flow patterns within the separator are independent of viscosity. 1 ET 0. 5 G(x) 0 Size x x50 xa xmax Fig. 4 Typical grade efficiency curve.
Features of G(x) curves: They may not start at the origin. x50 is that size of particle with an equal chance of reporting to either output stream. xmax is the maximum particle size and for x ? xmax G(x) = 1 (All larger particles are separated with 100% efficiency. xa is the analytical cut size and is that size which corresponds to ET = F(x ) where F(x) is the cumulative oversize distribution. x? is a cut size corresponding to the situation where the oversize cumulative distribution of the coarse product intersects the cumulative undersize distribution of the fines (or vice versa):- F (x? ) = 1 ? F (x? ) .
This value (x? ) is very sensitive to feed properties and operating conditions and has appears to have found little application outside of quality control. Why G(x) may not ? 0 as x ? 0 In solid liquid separators the flow liquids is divided between the overflow and underflow streams. Some fraction of the liquid reports to the underflow and carries with it a proportional fraction of available fine material (which effectively suffers no separation). This reduces the overall efficiency of the process. If the underflow to (feed) throughput ratio is Rf then the grade efficiency curve tends to a value of Rf as x ? 0.
The grade efficiency is effectively probabilistic in nature. The finite widths of inlet and outlets and the consequent range of particle trajectories within the separator only provide a chance of a particular particle following a particular trajectory. However, averaged over many particles (of identical size) the grade efficiency tends to a precise (probability) value. Limit of Separation There is always a value of particle size (x) above which G(x) = 1. This is the size (xmax) of the largest particle remaining in the overflow after separation. (That with an infinitely small chance of escaping. )
For example if a 100 i??m particle has no chance of escaping from the underflow of a settling process then neither with a particle of 105 i?? m (of the same material). Sharpness of cut The sharpness of cut (or separation) is related to the general slope of the grade efficiency curve, often expressed in terms of a ‘sharpness index’ which is defined in a variety of ways:- By the gradient of dG(x)/dx at x50 By the ratios of two sizes on the G(x) curve for which G(x) has prescribed values one either side of G(x) = 50%. (e. g. x25 /x75, x10 /x90 etc. or reciprocals thereof. ) Relationships between ET ,G(x), Fc(x), Ff(x) and F(x).
The grade efficiency function can be related to cumulative and frequency distributions:- (Mc)x Mc dFc/dx dFc G(x) = ?? = ???? = ET ??  (M)x M dF/dx dF This indicates how values of G(x) for various x can be obtained from the size distributions of feed and coarse product combined with a knowledge of the total efficiency ET. It is, however, necessary that all data be expressed using the same basis (number, surface, volume etc. ) and that the same analytical technique be employed to provide size distribution data for the various process streams and that those data be expressed in the same form.
Evaluation of G(x) can be performed graphically, or if size data are available as frequency distributions then direct automated numerical processing is possible. There is an advantage with the use of cumulative data. Graphical methods a) frequency data 1) Plot the distributions dF/dx and ET(dFc/dx) on the same diagram the ratio of the values of the ordinates of the two functions at any value of x give the corresponding value of G(x). (Numerical tables of frequency data are amenable to direct processing preferably by computer. ) b) cumulative data.
Many particle size analysers (Sedigraph, Coulter Counter etc. ) tend to produce cumulative and not frequency distributions. These too can be used to determine G(x) via a variety of data combinations (possible by rearrangement of the appropriate mass balance equations). In these cases the procedure is to plot (for example) Fc(x) as a function of F(x) for all available data points (in a square diagram) and then estimate the gradient (dFc/dF) at each data point on the resulting curve. Values of G(x) are then obtained by scaling the gradient of this curve by ET (as shown in the above equation).
Any procedure of this type is assisted by a knowledge of the asymptotes of the relationship between the chosen cumulative data distributions. A procedure involving evaluation of G(x) from ET, F(x) and Fc(x) (undersize) can exploit the fact that G(x) = 1 as x tends to ? and so dFc 1 ? = ? = G(? )  dF ET The line corresponding to this condition can be drawn on the plot, from the point (100, 100) to (100 ? ET ,0) (where ET is expressed as a percentage. )
Similarly when the grade efficiency is at a minimum (when x ? 0):- dFc Rf ? = ? = G(0)  dF ET,providing an asymptote through the points (0,0) and (100, Rf /ET). The curve through the (often scattered) data points should approach these asymptotes at extremes of particle size. They therefore provide a good guide to the ends of the G(x) curve. Similar constructions are possible using other combinations of data such as Ff(x), F(x) and ET. dFf(x) G(x) = 1 ? (1 ? ET) ??  dF(x) or  Although these equations can be used with either frequency distributions or cumulative types, in many situations use of the cumulative data leads to inherent accuracy.
Numerical differentiation of cumulative data leads to generation of errors – the fewer of such operations required by a procedure – the better. Cumulative data can be presented as oversize or undersize provided that consistent data sets are employed. Always retain the original form of size data to avoid repeated cycles of differentiation and summation(integration). In each of these situations asymptotic conditions (where G(x) ? Rf and G(x) = 1 serve to guide construction of curves relating the distributions.
The data combination which leads to an overall minimisation of random errors in both ET and G(x) is that of the overflow and the underflow (Ff(x) and Fc(x)). Evaluation of G(x) from Ff(x), F(x) and ET 1) Plot Ff (x) as a function of F(x) 2) A limiting line for this is the F(x) ? axis (Ff (x) = 0 so that G(x) = 1. 3) The other line is for the condition G(0) = Rf at x = 0 and has the slope of:- (1? Rf ) ??  (1? ET ) and passes through the appropriate corner of the diagram (where x ? 0). Estimated gradients from the curve are then scaled by a factor (1 ?
ET) and then subtracted from 1 to give G(x). Evaluation of G(x) from Ff(x), Fc(x) and ET 1) Plot Ff(x) as a function of Fc(x) 2) A limiting line is again the Fc axis (Ff(x) = 0) for x = ? ) and a line of slope ((1/Rf ) ? 1)/(1/(ET ? 1)/1) drawn through the point corresponding to x = 0 (in this case the top right hand corner). Evaluation of G(x) from Fc(x), Ff(x) and F(x) With analyses of input and product streams a priori knowledge of ET is unnecessary and can be evaluated i concert with G(x).
For this ET is eliminated from eqn. 12d using eqn. 15 leading to:- (F(x) ? Ff (x)) ?dFc(x) G(x) = ?????????  (Fc(x) ? Ff(x)) ? dF(x) The latter term (outside of brackets) can be evaluated at various points as previously described. A plot of the bracketed term against particle size provides estimates of ET from which some indication of the errors is available. Evaluation of ET from G(x) and F(x). If G(x) can be regarded as a characteristic function of a separator for particular conditions (dV/dt, ? etc. ) then ET can be estimated for a particular feed at the same conditions.
The size distribution of feed dF(x)/dx must be known then from eqn. 12d:- G(x)dF = ETdFc after integration over the size range 0 to xmax 1 1 ? G(x) dF = ET ? dFc [22a] 0 0 now as ET is constant and the integral on the RHS = 1, ET is given by ET = G(x) ? dF [22b] This can be evaluated either numerically or graphically. For the latter it is necessary to plot G(x) and F(x) against each other, for all available size data points, in a square diagram the entire area of which represents ET = 100% (i. e. G(x) = 100% for all x). The ratio of the area under the resulting curve to the total area provides a value of ET. 100 x? 0 Using scales as given ET = Area under curve/10000 G(x) % 0 0 100 F(x) %.
It is, possible to predict the size distribution of one of the products from those of the feed and other product in combination with the grade efficiency curve via the material balance relationships. Reduced Efficiency In separation systems with dilute and appreciable underflows the concept of ‘Reduced Efficiency’ is useful in describing the separation affect from any affects attributable to other causes, especially flow separation. The separator serves as a flow divider which results in some separation and thereby divides the solids too in at least a larger ratio as Rf (the ratio of underflow to throughput – which must be < 1).
The most widely adopted definition of ‘reduced’ total efficiency E’ is:- ET ? Rf E’ = ???  1 ? Rf The minimum efficiency due to ‘dead flux’ is therefore given by Rf. This equation gives a net efficiency of 0 when ET = Rf and complete separation when ET = 1. The grade efficiency also suffers from this problem (of divided flow) and can be correspondingly modified to give a reduced grade efficiency G'(x):- G(x) ? Rf G'(x) = ????  1 ? Rf This represents a net separation effect and, for inertial separation, goes through the origin since G'(0) = 0 (whereas G(0) = Rf ).
The particle size corresponding to G(x) = 0. 5 is termed the ‘reduced’ cut size x’. A concept widely used in descriptions of hydrocylone operations. Note the following: It still holds true that:-  but product size distributions cannot be calculated without a knowledge of Rf because equation  is inappropriate for G'(x) and E’. Combined Efficiencies Of Linked Separation Units It is advantageous to employ low cost dynamic separators as a first stage solids removal or classification step in order to improve the performance of an high efficiency (high cost) process further downstream.
It is necessary that the overall grade efficiency of the process be determined and this can be performed on an overall process basis or from G(x) functions for the individual units. Crawford M. (Air pollution control theory 1976 McGraw Hill) derives the following expression (and many others for a variety of arrangement):- G(x) = G1(x) + G2(x) ? G1(x)G2(x)  For n separators of efficiency G(x) in series this reduces to G(x) = 1? [1 ? G1(x)]n  which is subject to the law of diminishing returns! Particulate Systems 3 Particle Size Analysis and Grade Efficiency: page 1 of 27 ce321_2. doc.
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