Elementary separation factor

The elementary separation factor for a single-separating unit on a two component mixture Is defined as 

For the separation of uranium Isotopes by gaseous diffusion of UFg, the theoretical limiting value of a Is given by Graham s law 

The minimum reflux ratio required, at the feed point, to produce a given product rate P of material at isotopic composition Xp, in systems where a Is close to 1, Is 

This minimum feed/product rate would require an infinite number of separating stages in the cascade. 

Neither of these limiting cascade parameters are suitable for an actual isotope production task the minimum stage cascade 

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In a centrifugal field the energy per mole at a distance r from the axis is M(cor2)/2. M is molecular weight and go angular velocity. The ratio of partial pressures for two isotopomers (primed and unprimed) at the wall, rw, and on the axis, ro, yields the elementary separation factor. 

The Clusius-Dickel column is shown schematically in Figure 2. A wire is mounted at the axis of a cylinder. The wire is heated electrically and the outer wall is cooled. This sets up a radial thermal gradient which leads to a thermal diffusion separation in the x direction. As a result of the radial temperature gradient, a convection current is established in the gas, which causes the gas adjacent to the hot wire to move up the tube with respect to the gas near the cold wall. The countercurrent flow leads to a multiplication of the elementary separation factor. For gas consisting of elastic spheres, the light molecules will then concentrate at the top of the column, while the heavy molecules concentrate at the bottom. The transport theory of the column has been developed in detail (3, iS, 18) and will not be presented here. In a later section we shall discuss the general aspects of the multiplication of elementary separation processes by countercurrent flow. 

Our brief discussion of cascade principles serves to demonstrate the critical dependence of the size and operating costs of isotope separation plants on the elementary separation factor c. The size and initial cost are proportional to c 2. The operating cost is less sensitive to c, but varies at least as c The economic importance of these factors is readily seen in context with the separation of In 1960 the USAEC had three gaseous diffusion plants in operation. The cost of each plant was approximately 1 billion dollars the power consumption in each plant was 1,800,000 kw. If the plants were to be built with processes or equipment giving separation factors one half the one used, the additional construction cost to the U.S. taxpayers would be nine billion dollars. The increase in the annual operating costs of the plants can be conservatively estimated from the increase in the reflux ratio or power consumption to be 100,000,000/yr. This is a realistic demonstration of the economic benefits and importance of fundamental research and development to society. 

Eq. 9 Is of major Importance for estimating the size and cost of an Isotope separation plant. It Indicates that the total flow Is a product of two factors the first of these, proportional to l/(a-l) Is a function only of the elementary separation factor which Is determined by the separation process used. The second factor [In square brackets], which Is usually called the separative duty or separative work units (S.W.U.) Is a function only of quantities and concentrations of feed, product, emd waste. It has the same dimensions as those used for the quantities of material, and Its value Is Independent of the process used to accomplish the separation task. The significance of the magnitude of the elementary factor Is Immediately apparent a two-fold reduction In (a-1) requires an Increase In the total flow by a factor of 4. Since for a gaseous diffusion process, the total flow rate Is closely related to the total area of porous barriers, the total pumping capacity and the total power consumption required, all the associated costs vary proportionately. 

It is clear that the larger the elementary separation factor, e, and the number of equivalent theoretical plates, N, of the ion-exchange column are, the higher... 

K indicates how the two isotopes are distributed between the two chemical compoimds. Instead of K, very often the elementary separation factor (fi actionation factor) is used, which is defined by... 

The resulting temperature gradient establishes thermal diffusion in the radial direction and the consequent mass gradient causes convection. Molecules near the center move up the tube with respect to gas near the cold wall. The countercurrent flow thus established leads to multiplication of the elementary separation factor. The theory of such columns has been developed in detail, but will not be reviewed here (Jones and Furry 1946 London 1961). 

The elementary single stage separation factor a for a two component system is defined as... 

Other elementary criteria include the separation factor, S [2],... 

One method which can be used to establish the optimum conditions for the separation of a complex mixture (i.e., not only a pair) of compounds consists in searching for the maximum of a function denoted the chromatogram quality criterion. The evaluation of separation selectivity can be conducted with the aid of different criteria of chromatogram quality such as the sum of resolution, E 7 [6], the sum of separation factors, E 5 [2], and other sums and products of elementary criteria, selected examples of which are the resolution product, n Rs [7],... 

Both the separative power dU and the form of the value function V(x) can be determined by a single requirement that 5U ht independent of isotopic composition of the streams entering and leaving the unit. Subsequent analysis is considerably simplified (and yet remains sufficiently accurate for our purposes) if only close separation processes are considered. By close separation we mean that the elementary effect of each separating unit is quite small, or that the compositions of all streams entering and leaving the unit differ but little from each other. This condition is fulfilled if the separation factor a is close to unity. Under the close separation restraint, the value functions V Xp) and K(xw) may be obtained from Taylor series expansions about the feed composition ... 

Equations (7-29) and (7-32) both have the same form. It is easy to see that their temperature profiles are not linear. Their shapes are the same. Note that the temperature profile can be factored into two straight-line segments, one for each separate k. The composite will then be a line that curves upward in the usual plot. The tangent at any T can be used to obtain a value of an apparent activation enthalpy. The apparent activation enthalpy increases with temperature whenever the composite constant is a sum of the rate constants for elementary reactions. 

As a final remark it must be mentioned that theoretical and experimental works have been dedicated to investigating the effect of the finite size of the chains [65]. In fact, as grows exponentially, at low temperatures it can become comparable with the distance between two consecutive defects (e.g. impurities and vacancies) which are always present in real systems and hardly separated by more than 103 -104 elementary units. In case of Z < , the nucleation of the DW is energetically favoured if occurring at the boundaries, because the energy cost is halved. However the probability to have a boundary spin is inversely proportional to L thus the pre-exponential factor becomes linearly dependent on L, as experimentally found in doped SCMs. As doping occurs at random positions on the chain, a distribution of lengths is observed in a real system. However, as the relaxation time is only linearly dependent on L, a relatively narrow distribution is expected. 

While the NSE results show that, within the experimental accuracy, in the range Q<0.15 A the Rouse model gives a good account for the internal modes as well as for the diffusion of the chain centre of mass, it is also clear that for higher Q-values the experimental structure factors decay significantly more slowly than the Rouse model would require. These deviations are quantified in fitting the Rouse model to the different spectra separately. This procedure results in a strong dispersion of the elementary Rouse rate. The values determined for at Q>0.15 A" follow a Q-dependence, which can be described by the power law ... 

When there are only two electrons the analysis is much simplified. Even quite elementary textbooks discuss two-electron systems. The simplicity is a consequence of the general nature of what is called the spin-degeneracy problem, which we describe in Chapters 4 and 5. For now we write the total solution for the ESE 4 (1, 2), where the labels 1 and 2 refer to the coordinates (space and spin) of the two electrons. Since the ESE has no reference at all to spin, 4 (1, 2) may be factored into separate spatial and spin functions. For two electrons one has the familiar result that the spin functions are of either the singlet or triplet type. 

The transition-state approach permits us to make a separation of the factors constituting an experimental specific rate constant (for an elementary chemical act) into kinetic and thermodynamic factors. Thus, for the transition state X = A + B + C+ we can write for the rate constant jfc governing the appearance of products (Sec. XII.4) from X... 

The available reliable information on the rate coefficient of reaction (xvi) depends almost entirely on fast flow-discharge studies, and, with the exception of one recent shock tube result, direct measurements are confined to near 300 K. Even here there is a factor of two or three disagreement between authors. Results are summarized in Table 39. Uncertainties arise from two major causes. First, the second order gas phase decay of OH is accompanied by a first order heterogeneous decay. Optimization of the separation of the observed decay into its first and second order components is difficult, and this may account for some of the reported discrepancies [222]. Secondly, all the investigations have used the H + NO2 reaction as the source of OH, with the NO2 added downstream of the discharge. The relevant elementary steps causing the growth and decay of OH are then... 

The thermal diffusion factor a is proportional to the mass difference, (mi — mo)/(mi + m2). The thermal diffusion process depends on the transport of momentum in collisions between unlike molecules. The momentum transport vanishes for Maxwellian molecules, particles which repel one another with a force which falls off as the inverse fifth power of the distance between them. If the repulsive force between the molecules falls off more rapidly than the fifth power of the distance, then the light molecule will concentrate in the high temperature region of the space, while the heavy molecule concentrates in the cold temperature region. When the force law falls off less rapidly than the fifth power of the distance, then the thermal diffusion separation occurs in the opposite sense. The theory of the thermal diffusion factor a is as yet incomplete even for classical molecules. A summary of the theory has been given by Jones and Furry 15) and by Hirschfelder, Curtiss, and Bird 14), Since the thermal diffusion factor a for isotope mixtures is small, of the order of 10", it remained for Clusius and Dickel (8) to develop an elegant countercurrent system which could multiply the elementary effect. 

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