Partition cycle

The Purex Process has been used either with "early" or with "late" Pu-U partitioning (Fig. 1). The early split in the first cycle avoids the co-stripping of uranium and plutonium which may result in higher plutonium losses. In other flowsheets, the partitioning is delayed until the second cycle. While no particular trend has been obvious, some of the recent flowsheets in which the plutonium is reduced prior to re-extraction (10,11) will necessarily require a second cycle partitioning. 

Comparison. Pick two sets of data from print or elecU onic sources, preferably from the same field (e.g., health, finance, sociology). Compare the sets in terms of how well the organization of the data enables you to derive the intended conclusions. In your comparison, you may want to apply the terms introduced in this section (permutations, cycles, partitions, etc.) and in previous chapters and sections. 

Remark (The conjugacy classes of symmetric groups) We recall that two elements 7T and a of S are conjugate if and only if they have the same cycle partition, or, in other words, if and only if they are of the same cycle type ... 

Finally, each partition a h n occurs as the cycle partition of a permutation n in S , and hence for the cycle types a H n, too. 

These graphs are orbits of the symmetric group S4 which acts on the set 4 = 0,1,2,3. Its conjugacy classes correspond to the different cycle partitions or cycle types,... 

We list a transversal of the conjugacy classes together with the corresponding cycle partition, the cycle type and the order of the class considered ... 

There are many other applications. They include detemiination of the ratios of the partition coefficients P IPq) of solutes B and C in two different solvents by using the themiodynamic cycle ... 

Chemical Separation. A reprocessing facility typically utilizes multiple extraction/reextraction (stripping) cycles for the recovery and purification of uranium and plutonium. For example, a co-decontamination and partitioning cycle is followed by one or more cycles of uranium and plutonium purification. The basic process is illustrated in Figure 3. 

To answer the question one has to examine carefully the permutations which correspond to the 24 rotations of the octahedron. We partition these permutations into cycles and assign to each cycle of a certain order k the symbol f. assign to a cycle of order 1 (vertex which is invariant under rotation), f to a cycle of order two (transposition), /g to a cycle of order three, etc. A permutation which is decomposed into the product of cycles with no common elements is represented by the product of the symbols /. associated with the corresponding cycles. Thus the rotations of the octahedron are described by the following products ... 

It would take us too far out of our way to explore the territory opened up by this way of looking at Polya s Theorem, but we can take a quick look at the view and pick out some of the salient features. The cycle index, in its new clothes, is expressible in terms of -functions conversely, an 5-function can be regarded as a sort of cycle index. The 5-function for a partition can be defined by... 

The use of Polya s Theorem in a specialized context such as the above, has led to the extension of the theorem along certain useful lines. One such derivation pertains to the situation where the boxes are not all filled from the same store of figures. More specifically, the boxes are partitioned into a number of subsets, and there is a store of figures peculiar to each subset. To make sense of this we must assume that no two boxes in different subsets are in the same orbit of the group in question. A simple extension of Polya s Theorem enables us to tackle problems of this type. Instead of the cycle index being a function of a single family of variables, the 5j, we have other families of variables, one for each subset. An example from chemical enumeration will make this clear. 

In a cycle of operation the liquid enters port A and fills the spaces 1 and 3, thus forcing the piston to oscillate counterclockwise opening spaces 2 and 4 to port B. Because of the partition, the piston moves downwards so that space 3 is cut off from port A and becomes space 4. Further movement allows the exit port to be uncovered, and the measured volume between hub and piston is then discharged. The outer space 1 increases until the piston moves upwards over the partition and space 1 becomes space 2 when a second metered volume is discharged by the filling of the inner space 3. Meters of this type will handle flows of between about 0.005 and 15 litres/s. 

Figure 1. Global Environment Cycles and Spatial Partitions.

Chemically reactive elements should have a short residence time in seawater and a low concentration. A positive correlation exists between the mean ocean residence time and the mean oceanic concentration however, the scatter is too great for the plot to be used for predictive purposes. Whitfield and Turner (1979) and Whitfield (1979) have shown that a more important correlation exists between residence time and a measure of the partitioning of the elements between the ocean and crustal rocks. The rationale behind this approach is that the oceanic concentrations have been roughly constant, while the elements in crustal rocks have cycled through the oceans. This partitioning of the elements may reflect the long-term chemical controls. The relationship can be summarized by an equation of the form... 

Although the largest reservoirs of carbon are found in the lithosphere, the fluxes between it and the atmosphere, hydrosphere, and biosphere are small. It follows that the turnover time of carbon in the lithosphere is many orders of magnitude longer than the turnover times in any of the other reservoirs. Many of the current modeling efforts studying the partitioning of fossil fuel carbon between different reservoirs only include the three "fast" spheres the lithosphere s role in the carbon cycle has received less attention. 

Fig. 12-2 Partitioning of the various forms of nitrogen in the atmosphere. Units are Tg N. (Reprinted with permission from R. Soderlund and T. Rosswall, The nitrogen cycles. In O. Huntizger (1982). "The Natural Environment and the Biogeochemical Cycles," p. 70, Springer-Verlag, Heidelberg.)...

Figure 13-5 is the box model of the remote marine sulfur cycle that results from these assumptions. Many different data sets are displayed (and compared) as follows. Each box shows a measured concentration and an estimated residence time for a particular species. Fluxes adjoining a box are calculated from these two pieces of information using the simple formula, S-M/x. The flux of DMS out of the ocean surface and of nss-SOl back to the ocean surface are also quantities estimated from measurements. These are converted from surface to volume fluxes (i.e., from /ig S/(m h) to ng S/(m h)) by assuming the effective scale height of the atmosphere is 2.5 km (which corresponds to a reasonable thickness of the marine planetary boundary layer, within which most precipitation and sulfur cycling should take place). Finally, other data are used to estimate the factors for partitioning oxidized DMS between the MSA and SO2 boxes, for SO2 between dry deposition and oxidation to sulfate, and for nss-SO4 between wet and dry deposition. 

Ketogenesis is regulated at three cmcial steps (1) control of free fatty acid mobihzation from adipose tissue (2) the activity of carnitine palmitoyltransferase-1 in hver, which determines the proportion of the fatty acid flux that is oxidized rather than esteri-fied and (3) partition of acetyl-CoA between the pathway of ketogenesis and the citric acid cycle. 

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