Cascade total flow rates

The second factor appearing in Eqs. (12.122) and (12.139) for the total flow rate in an ideal cascade is known as the separative capacity, or separative power [C3], D. For a plant with a single tails, product, and feed stream, it is given by... 

The importance of the separative capacity in isotope separation lies in the fact that it is a good measure of the magnitude of an isotope separation job. Many of the characteristics of the plant that make important contributions to its cost are proportional to the separative capacity. For example, in a gaseous diffusion plant built as an ideal cascade of stages operated at the same conditions, the total flow rate, the total ptimp capacity, the total power demand, and the total barrier area are all proportional to the separative capacity. In a distillation plant, the total column volume and total rate of loss of availability are proportional to the separative capacity. 

The total inventory of the enriching section Ip then is just h times the total flow rate in the enriching section for a close-separation, ideal cascade. 

Figure 13.8 compares the variation of tails flow rate with stage number in a squared-off cascade with the variation in an ideal cascade performing the same job of separation in the same number of stages. Because the total flow rate in an ideal cascade is the lowest possible, the area under the stepped curve of the squared-off cascade is greater than under the smoothly tapered curve of the ideal cascade. 

Analysis of fractional extraction is straightforward for dilute mixtures when the solutes are independent and total flow rates in each section are constant. The external mass balances for the fractional extraction cascade shown in Figure 13-5 are... 

The other method for collecting small particles, and the method used in the MOUDl cascade impactor, is to employ very small nozzles as described by Kuhlmey et al. (1981). The small dimensions of the nozzle allow for small particles to be collected at relatively low jet velocities, and consequently low pressure drops. By the use of multiple micro-orifice nozzles in the MOUDl cascade impactor (2000 nozzles of 52 pm in diameter in the final stage), the cut-size of the final stage can be as low as 0.056 pm with a total flow rate through the impactor of 30Lmin ... 

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 sum of the stage feed flow rates of all of the stages in an ideal cascade is just twice the total cascade upflow rate when (a — 1) is small with respect to unity, or... 

The fact that the total internal flow rate in a close-separation, ideal cascade is given by Eq. (12.142) may be derived without solving explicitly for the individual internal flow rates by the following development, due originally to P. A. M. Dirac. This procedure is valuable in showing the fundamental character of the separation potential and the separative capacity, and provides a point of departure for the treatment of multicomponent isotope separation. 

We wish to determine the effect of using a cascade of two CSTRs that differ in size on the volume requirements for the reactor network. In Illustration 8.8 we saw that for reactors of equal size, the total volume requirement was 6.72 m. If the same feed composition and flow rate as in Illustration 8.8 are employed and if the reactors are operated isothermally at 25°C, determine the minimum total volume required and the manner in which the volume should be distributed between the two reactors. An overall conversion of 0.875 is to be achieved. 

A schematic of a countercurrent leaching system is shown in Figure 14-4 with the appropriate nomenclature. Since leaching is quite similar to LLE, the same nomenclature is used (see Table 13-2L Even if flow rates E and R vary, it is easy to show that the differences in total and conponent flow rates for passing streams are constant. Thus, we can define the difference point from these differences. This was done in Eqs. fl3-42i and fl3-43i for the LLE cascade of Figure 13-20. Since the cascades are the same (compare Figures 13-20 and 14-4L the results for leaching are the same as for LLE. 

What production capacity of C could be obtained in a reactor cascade consisting of three identical CSTRs in series, with a total volume of 500 dm The volumetric flow rate and initial concentrations are the same as in case b. 

Now employ the result (9.1.6) for N to develop the following expression for the total light fraction flow rate (upflow) from stage 1 to IV in the enriching section of an ideal cascade ... 

Obtain an expression for the total heavy fraction flow rate (downffow) in the enriching section of the cascade, i.e. develop a continuous version of Problem 9.1.1 for an illustration of the... 

Determine the mass rate flow of the top cycle, power required by compressors 1 and 2, total power required by the compressors, rate of heat added to the evaporator, cooling load, and COP of the cascade vapor refrigeration cycle. 

The interface between continuous controls and sequence logic (discussed shortly) is also important. For example, a feed might be metered into a reactor at a variable rate, depending on another feed or possibly on reactor temperature. However, the product recipe calls for a specified quantity of this feed. The flow must be totalized (i.e., integrated), and when the flow total attains a specified value, the feed must be terminated. The sequence logic must have access to operational parameters such as controller modes. That is, the sequence logic must be able to switch a controller to manual, automatic, or cascade. Furthermore, the sequence logic must be able to force the controller output to a specified value. 

The distance the spray extends away from the tank wall is assumed to be 1.5 m over the full height of the cascade. This is a reasonable minimum figure based on observations on water cascades. Wind girders part way down the tank can increase the width to in excess of 3 m but any broadening of the liquid cascade increases the total induced air flow and tends to reduce the maximum vapour concentration. Given the cross section of the cascade and the total liquid release rate the liquid mass density can be calculated. 

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