Cut size analytical

Particle sizes dso and da, corresponding to G(d) = 0.5 and G(d) = E, serve as useful parameters of the function. The former indicates the size of particle with an equal probability of reporting to either outlet. The latter is usually termed the analytical cut size and corresponds to a cumulative (oversize) frequency of the feed material which divides it exactly in the proportion of E, as though GW) were a step function. 

Two other cut sizes are also used [4] The analytical cut size, x is that at which the feed is split in proportions given by the total efficiency. This definition implies that the amount of displaced material in the coarse stream is balanced by the amount of displaced coarse material in the fine stream. This is also the condition that the analytical cut size is equal to the equiprobable size. As this rarely happens in practice the two sizes are usually different. The analytical cut size is less useful than the... 

Intersection cut size Xp is defined as the size at which the cumulative percentage oversize of the coarse stream is equal to the cumulative percentage undersize of the fine stream. This size requires just two size determinations on samples from the two streams it is however even more sensitive to changes in the feed size distribution than the analytical cut size. The grade efficiency is often expressed as a single number. This number is known as the sharpness index, y/, and is a measure of the slope of the grade efficiency curve ... 

Elutriation and fluid classification methods are also highly relevant to solid-liquid separation problems—they use the same or similar mechanisms for analysis as many separators. Use is made here of the size-dependent nature of dynamic separation processes and most of these methods are based on the analytical cut size defined in chapter 3, Efficiency of Separation . 

The so-called analytical cut size Xa is the size such that ideally the feed solids would be split according to size (with no misplaced material) in the proportions given by the total efficiency Ej. In other words the analytical cut size corresponds to the percentage equal to on the cumulative particle size distribution oversize F x) of the feed material (see Figure 3.5a), i.e. 

The values of the other two commonly used cut sizes, the analytical cut size Xa and the curve intersection cut size Xp (as defined in section 3.2.2.1.1) are always different from X50, unless X50 is equal to the median of the feed size distribution Xg when all three coincide the difference between X50, and and Xp increases as the median of the feed moves towards either end of the grade efficiency curve. This is best shown in the plot of G(x) against F(x) in Figure 3.13, from which it can be seen that both x and Xp are always on the same side of X50 as the median Xg of the feed, with x being always closest to X50, hence ... 

Theoretical conversion of the easily obtainable analytical cut size Xa into the equiprobable size X50 is possible if both the feed size distribution and the grade efficiency curve can be approximated by an analytical function. Thus for example if both of the above-mentioned functions are log-normal it can be shown that the total efficiency Ej can be determined analytically from the following expression ... 

By definition of the analytical cut size Xa the cumulative percentage oversize of the feed at Xa is equal to the total efficiency Ei (equation 3.13) and hence, by integration of equation 2.21,... 

Due to the fact that the grade efficiency curves are not ideal that is do not form a step function, which jumps from zero to one at a cenain 5ft>ke -number, the corresponding point of the settling rate distribution curve can only be determined cor-recdy if the amounts of misplaced material in the fine and the coarse fracnon are the same. Fig. 8 shows the situation. The amount collected on the plate is represented by the bell shaped curve on the right for the coarse fraction. It represents the residue. The vertical line must be shifted to a point, x, until the areas I and n, are equal. The residue measured on the plate can then be plotted with respect to x, the analytical cut size /29/. The me-... 

Since tbe grade efficiency function G is needed to know tbe equiprobable size tp so, and substantial information is necessary before G is known (e.g. ErtfzO p)) and jy (tp) should be available according to equation (2.4.4b)), two other cut-size definitions are fi equently used in industry the analytical cut size Tp and the cut size by curve intersection (Svarovsky, 1979). We will only touch upon the analytical cut size here. 

The analytical cut size Tp is defined such that a hypothetical and ideal screen having this size opening will give from the feed solid mixture the same value of total efficiency Et as the actual separator. In terms of the feed particle size distribution fimction, ly(rp). 

The analytical cut size Tp,p is not equal to the equiprobable cut size tp,5o. The relation between these two can be derived, for example, for a log-normal particle size distribution. 

Figure 20. Cycling behavior of Sn+SnSb powder (analytical composition "SnassSb(U2particle size <0,2pm) and Sn powder (particle size

Until this point, the sample preparation techniques under discussion have relied upon differences in polarity to separate the analyte and the sample matrix in contrast, ultraflltration and on-line dialysis rely upon differences in molecular size between the analyte and matrix components to effect a separation. In ultrafiltration, a centrifugal force is applied across a membrane filter which has a molecular weight cut-off intended to isolate the analyte from larger matrix components. Furusawa incorporated an ultrafiltration step into his separation of sulfadimethoxine from chicken tissue extracts. Some cleanup of the sample extract may be necessary prior to ultrafiltration, or the ultrafiltration membranes can become clogged and ineffective. Also, one must ensure that the choice of membrane filter for ultrafiltration is appropriate in terms of both the molecular weight cut-off and compatibility with the extraction solvent used. 

A 0.5-2-kg crop sample is cut into small pieces and homogenized thoroughly using a food processor. Rice grain is milled with an ultracentrifuge mill and sieved through a 42-mesh screen. The typical size of an analytical sample is less than 50 g. To prevent the decomposition of the anilide residues, crop samples should be frozen soon after collection and maintained frozen until analyzed. 

Nondestructive radiation techniques can be used, whereby the sample is probed as it is being produced or delivered. However, the sample material is not always the appropriate shape or size, and therefore has to be cut, melted, pressed or milled. These handling procedures introduce similar problems to those mentioned before, including that of sample homogeneity. This problem arises from the fact that, in practice, only small portions of the material can be irradiated. Typical nondestructive analytical techniques are XRF, NAA and PIXE microdestructive methods are arc and spark source techniques, glow discharge and various laser ablation/desorption-based methods. On the other hand, direct solid sampling techniques are also not without problems. Most suffer from matrix effects. There are several methods in use to correct for or overcome matrix effects ... 

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