Cascade separative capacity

Application. In addition to providing a relatively simple means for estimating the production of separation cascades, the separative capacity is useful for solving some basic cascade design problems for example, the problem of determining the optimum size of the stripping section. 

It can be assumed that P,Jp, and for the cascade have been specified, and that the cost of feed and the cost per unit of separative work, the product of separative capacity and time, are known. The basic assumption is that the unit cost of separative work remains essentially constant for small changes ia the total plant size. The cost of the operation can then be expressed as the sum of the feed cost and cost of separative work ... 

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... 

The first factor is a measure of the relative ease or difficulty of the separation it is large when is close to unity and small when differs markedly from unity. The second factor is a measure of the magnitude of the job of separation it is proportional to the throughput, and is large when product and tails differ substantially in composition from feed, and small when these compositions are nearly equal. The second factor has been termed the separative capacity, because it is a measure of the rate at which a cascade performs separation. It equals the sum of two output terms, each the product of an output flow rate and a function of the corresponding output condition, minus an input term that is the product of the feed rate and a function of the input condition. The separative capacity is discussed in more detail in Sec. 10. 

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 separative capacity has the same dimensions as used for the flow rates. It is a measure of the rate at which a cascade is performing separation. 

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 separative capacity is analogous to the heat duty of an evaporator or other process equipment. The separation potential is analogous to the enthalpy per mole of the streams entering or leaving an evaporator. Calculations of material balances and separative capacity in an isotope separation plant are made in similar fashion to conventional material and heat balances. A form for such calculations is illustrated in Table 12.8, which illustrates the calculation of the separative capacity of an isotope separation cascade producing 1 mol/day of at 0.80 mole... 

It is also useful to have a measure of the amount of separation performed by a cascade in making Ep moles of product and E / moles of waste from Ep moles of feed. This measure is provided by the separative work S, defined in similar fashion to the separative capacity. 

If the rate of production of an ideal cascade at one set of feed, product, and tails compositions is known, so that its separative capacity can be evaluated, the best possible performance of the cascade for another set of compositions can be calculated by treating its separative capacity as a constant property of the cascade. This will be true if under the changed conditions the number of separating units in series and parallel are so rearranged that mixing of streams of different compositions is avoided. 

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 consider a close-separation, ideal cascade Wiose external streams have molar flow rates Xif (positive if a product, negative if a feed), and compositions X/t expressed as mole fraction. Let us look for a function of composition x ), to be called the separation potential, with the property that the sum over all external streams, to be called the separative capacity D,... 

Figure 12.22 represents stages —2, i—1, i, and / + 1 of such a cascade, with the th product stream consisting of part of the heads stream of stage / — 1. The total internal flow from stage i is Mi + A /- The separative capacity of the rth stage, considered as an isolated plant, is... 

When the separation potential satisfies (12.172), the separative capacity of a single stage in a close-separation cascade operated at a cut of (M = 1V) from Eq. (12.170) is... 

We shall now show that the separative capacity of the entire cascade, D, is given by... 

Thus, we have shown that the separative capacity of an ideal cascade is the sum of the separative capacities of its component stages. And if the separation potential satisfies the differential equation (12.172), the total internal flow is given by... 

When interstage flows are adjusted so that the abundance ratios R of to of each pair of streams being mixed are equal, the separative capacity D of an entire cascade whose feed, product, and tails are... 

Assume that the plant can be rearranged to operate as an ideal cascade at constant separative capacity when operating conditions are changed individually as described in each of the following ways ... 

Consider a cascade to produce 1.25 mol/day of uranium enriched to 80 m/o from natural uranium feed containing 0.72 m/o while stripping tails to 0.36 m/o. The separative capacity of such a cascade was evaluated in Table 12.8. 

Stage separative capacity. The separative capacity A of a stage in a close-separation cascade with... 

After the cascade improvement and cascade operating programs planned 6y U.S. DOE have been completed, their power consumption will be increased to 7,380,000 kW and their separative capacity to 27,700,000 kg SWU/year, equivalent to a specific power consumption of 0.266 kW/(kg SWU/year). 

The application of cascade isotachophoresis permits to increase the separation capacity by chemical means, the detection conditions lemainmg unaifected. 

According to Eq. (15), the number of separating units in a cascade of specified total separative capacity U is minimized if the separative power of each unit is maximized. Consequently, we determine values of L, m, and 0 which maximize 6U. This optimization procedure is different from that discussed in Section II,H. In the latter, 9 and L were fixed and m was chosen to maximize a. In the present case, we shall fix m and choose 9 and L to maximize dU. Optimization of the controllable variables to give the largest separative power is more important than adjustments to achieve the largest separation factor dU determines the number of centrifuges required in a cascade, whereas a merely determines how the fixed number of units is to be arranged in the ideal cascade. 

Mechanical methods also exist for dividing up particulate material into suitably sized samples. Samples obtained by these means are usually representative of the bulk material within limits of less than 1 per cent, and are based upon the requirements established by the British Standards Institution. Sample dividers exist with capacities of up to 10 L and operate either by means of a series of rapidly rotating sample jars under the outlet of a loading funnel, or by a rotary cascade from which the samples are fed into a series of separate compartments. Sample dividers can lead to a great deal of time-saving in laboratories dealing with bulk quantities of powders or minerals. 

A comparison for the different cases of the production costs and dependence on the annual capacity is given in Fig. 8.1-4. The results are based on the cascade-operation mode, with three extractors, extraction at 280 bar and 65°C, cycle-times of 7.5 hours, and a separation pressure of 60 bar for the non-isobaric process. 

Another decision for the design engineer is the selection of the operation mode, whereby he can choose between the single- and cascade-mode of operation. In principle this is valid for multipurpose plants of medium scale, such as plants to extract spices and/or herbs. Fig. 8.1-6 gives data on the different production costs for a plant with a total extraction volume of 600 1, operated in different modes, but with the same capacity. The investigation is based on equal batch times (in our case 4 hours), equal mass-flow per kg of raw material and, of course, equal extraction- and separation conditions. 

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