Cascade equations

Dienylacetates give bicyclo[3.3.0]octanones in reasonable yields and diastereoselectivity, depending on conditions, temperature, and pressure, though the main product of these reactions often results, from premature termination of cascade (Equation (38))/ ... 

Palladium chemistry has been used in the synthesis of tetrahydroisoquinolines. Different combinations of iodoaryl-amine-alkene can be used in these multicomponent reactions. For example, the metal-mediated o-alkylated/alkenyl-ation and intramolecular aza-Michael reaction (Scheme 109) give moderate yields of heterocycle <2004TL6903>, whereas the palladium-catalyzed allene insertion-nucleophilic incorporation-Michael addition cascade (Equation 172) produces good yields of tetrahydroisoquinolines in 15 examples <2003TL7445> with further examples producing tetrahydroquinolines (Scheme 110) <2000TL7125>. 

The overall separation depends on the single stage separation factor, the number of separating elements, and design and operating characteristics of the cascade. Equations 5-8 are insufficient to determine all the variables. It is instructive to consider three types of cascades the minimum stage cascade, the minimum reflux cascade, and the ideal cascade. The material balance equations from the i + 1 th stage to the product of the cascade lead to... 

Production of secondary cosmic rays and 7-rays in the interstellar medium generally involves less than one interaction per primary. In the language of accelerators, this is the thin-target regime. In contrast, the depth of the atmosphere is more than ten hadronic interaction lengths, so we have a thick target to deal with. The relevant cascade equation is... 

The same set of cascade equations (see Eq. 19) governs air showers and uncorrelated fluxes of particles in the atmosphere. The boundary condition for an air shower initiated by a primary of mass A and total energy Eq is... 

Uncorrelated fluxes in the atmosphere. The simplest physical example to illustrate the solution of Eq. 22 is to calculate the vertical spectrum of nucleons as a function of depth in the atmosphere. Nucleons are stable compared to the transit time through the atmosphere, so only losses due to interactions are important in the cascade equation 19. In the approximation of scaling, the dimensionless distribution Fn n(F, E ) —> F( ) as in Eq. 9, with = E/E. Eq. 19 becomes... 

We can also And the weight fraction distribution function Wm(a) from the cascade equations (3.84a) and (3.84b) by expanding the function Fq(x) in powers of the dummy parameter 0. This procedure is easily feasible if we apply the following Lagrange theorem [22,23]. The theorem states that if the variable x is related to 9 by the equation... 

Dual solvent fractional extraction (Fig. 7b) makes use of the selectivity of two solvents (A and B) with respect to consolute components C and D, as defined in equation 7. The two solvents enter the extractor at opposite ends of the cascade and the two consolute components enter at some point within the cascade. Solvent recovery is usually an important feature of dual solvent fractional extraction and provision may also be made for reflux of part of the product streams containing C or D. Simplified graphical and analytical procedures for calculation of stages for dual solvent extraction are available (5) for the cases where is constant and the two solvents A and B are not significantly miscible. In general, the accurate calculation of stages is time-consuming (28) but a computer technique has been developed (56). 

The defects generated in ion—soHd interactions influence the kinetic processes that occur both inside and outside the cascade volume. At times long after the cascade lifetime (t > 10 s), the remaining vacancy—interstitial pairs can contribute to atomic diffusion processes. This process, commonly called radiation enhanced diffusion (RED), can be described by rate equations and an analytical approach (27). Within the cascade itself, under conditions of high defect densities, local energy depositions exceed 1 eV/atom and local kinetic processes can be described on the basis of ahquid-like diffusion formalism (28,29). 

In the design of cascades, a tabulation of p x) and of p (x) is useful. The solution of the above differential equation contains two arbitrary constants. A simple form of this solution results when the constants are evaluated from the boundary conditions u(0.5) = u (0.5) = 0. The expression for the value function is then ... 

For the case under consideration, where the value of a — 1 is quite small, it follows that everywhere in the cascade, except possibly at the extreme ends, the stage upflow is many times greater than the product withdrawal rate. Thus L/ L — P) can be set equal to unity. Furthermore when the value of a — 1 is small, the stage enrichment can be approximated by the differential ratio dx/dn without appreciable error. The gradient equation for the... 

Equations 27 and 28 can be used in conjunction, along with the corresponding equations for the stripping section, to produce an ideal plant profile such as is shown in Figure 4 where F is plotted against for the example of an ideal cascade to produce one mol of uranium per unit time enriched to 90... 

For the ideal cascade, T is given by equation 28 and dx/dn by equation 26. Making these substitutions ... 

The total flow in the cascade is then given by the sum of equations 32 and 33, which can be simplified with the use of the cascade material balances ... 

The second term in brackets in equation 36 is the separative work produced per unit time, called the separative capacity of the cascade. It is a function only of the rates and concentrations of the separation task being performed, and its value can be calculated quite easily from a value balance about the cascade. The separative capacity, sometimes called the separative power, is a defined mathematical quantity. Its usefulness arises from the fact that it is directly proportional to the total flow in the cascade and, therefore, directly proportional to the amount of equipment required for the cascade, the power requirement of the cascade, and the cost of the cascade. The separative capacity can be calculated using either molar flows and mol fractions or mass flows and weight fractions. The common unit for measuring separative work is the separative work unit (SWU) which is obtained when the flows are measured in kilograms of uranium and the concentrations in weight fractions. 

The great utility of the separative capacity concept Hes in the fact that if the separative capacity of a single separation element can be deterrnined, perhaps from equations 7 or 10, then the total number of such identical elements required in an ideal cascade to perform a desired separation job is simply the ratio of the separative capacity of the cascade to that of the element. The concept of an ideal plant is useful because moderate departures from ideaUty do not appreciably affect the results. For example, if the upflow in a cascade is everywhere a factor of m times the ideal upflow, the actual total upflow... 

Time-Dependent Cascade Behavior. The period of time during which a cascade must be operated from start-up until the desired product material can be withdrawn is called the equiUbrium time of the cascade. The equiUbrium time of cascades utilizing processes having small values of a — 1 is a very important quantity. Often a cascade may prove to be quite impractical because of an excessively long equiUbrium time. An estimate of the equihbrium time of a cascade can be obtained from the ratio of the enriched inventory of desired component at steady state, JT, to the average net upward transport of desired component over the entire transient period from start-up to steady state, T . In equation form this definition can be written as... 

Equations 43 and 44 thus yield a lower and upper limit, respectively, and used together usually give a satisfactory estimate for the equihbrium time of a cascade. 

Examination of equation 42 shows that T is directly proportional to the average stage holdup of process material. Thus, in conjunction with the fact that hquid densities are on the order of a thousand times larger than gas densities at normal conditions, the reason for the widespread use of gas-phase processes in preference to hquid-phase processes in cascades for achieving difficult separations becomes clear. 

The actual power requirement is greater than that given by equation 58 or 60 because of the occurrence of frictional losses ia the cascade piping, compressor iaefftciencies, and losses ia the power distribution system. 

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