Equiprobable size

The 50% size on the grade efficiency curve is called the equiprobable size since particles of this size have an equal chance of being in either the coarse or the fine stream for this example e = 16.7 pm. Figure 5.1a shows how the feed is split between the coarse and fine fraction i.e. Wfipc) = W/J,x)+W/pc). 

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... 

If the complex separation performance of a size-dependent separator (best shown by a full grade efficiency curve) has for some reason to be described by a single number, in most applications it is best to use the concept of a cut size. In such a case, the cut size has to be unique and, if possible, independent of the size distribution of the feed solids, so that it characterizes only the machine run under the chosen conditions of operation. As the whole of the grade efficiency curve usually satisfies this condition, it is best to define the cut size as corresponding to a unique point on the curve. The equiprobable size X50 is probably best suited for this purpose and it is certainly the best of the three definitions of cut size reviewed here (section 3.2.2.1). 

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 ... 

X50 and fTgs are the equiprobable size (cut size) and the geometric standard deviation of the grade efficiency function respectively, and the erf function is defined as... 

Several authors have attempted to calculate particle trajectories in the cyclone and to derive formulae at least for the equiprobable size X50 if not for the whole grade efficiency curve. Some of these theories are discussed in section 6.6 certain observations are given here and refer to the probable behaviour of solid particles in sufficiently dilute suspensions. 

For such a curve, the particle size for which the grade efficiency is 50%, X50, is often used as a single number measurement of the efficiency of the cyclone. X50 is also know as the equiprobable size since it is that size of particle which has a 50% probability of appearing in the coarse product. This also means that, in a large population of particles, 50% of the particles of this size will appear in the coarse product. X50 is sometimes simply referred to as the cut size of the cyclone (or other separation device). 

One of the most commonly used cut sizes is the equiprobable size Cp so, for which the value of the grade efficiency is 0.50 (Figure 2.4.4). A particle having this size has an equal probability of appearing in both the overflow, as well as the underflow from the separator. A smaller particle will most likely be carried away by the fluid in the overflow whereas a larger particle will most likely be separated from tbe fluid and appear in the underflow (Svarovsl, 1979). Note that if G is plotted against a normalized particle radius (tp the equiprobable size will be independent... 

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. 

For reven sible systems, evolution almost always leads to an increase in entropy. The evolution of irreversible systems, one the other hand, typically results in a decrease in entropy. Figures 3.26 and 3.27 show the time evolution of the average entropy for elementary rules R32 (class cl) and R122 (class c3) for an ensemble of size = 10 CA starting with an equiprobable ensemble. We see that the entropy decreases with time in both cases, reaching a steady-state value after a transient period. This dc crease is a direct reflection of the irreversibility of the given rules,... 

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... 

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. 

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