Minimum separation work

An average value of the differential enthalpy Ahj of the phase change may be assumed, by integrating Eq. (1-139) in the temperature interval T] - Tj. This may be done, if no measured data is available, and the shape of the equilibrium 2 ,(7 is known, using 

The separation factor is therefore a direct measure of the separation efficiency of a separation unit or the whole process, and is thus of practical importance. It is usually dependent on pressure, temperature, and phase composition. Large values for the separation factor characterize a process whith a low separation effort. The more a approaches unity, the more difficult the separation. When a = 1 separation is impossible. 

The separation factor a is the relative volatility in distillation processes. The light phase is the vapor phase and the heavy phase is the liquid phase the component with the lower boiling point is the light component. Taking Dalton s and Raoult s laws into account, Eq. (1-142), the relative volatility of the lighter component 1 referred to the heavier component 2, becomes 

Substituting the mole concentration X and T, with the mole fraction and (see Table 1-4) the separation factor a, 2 is 

In this case, is dependent only on pressure and temperature and not on the composition of the mixture. 

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When this is combined with the definition of minimum separation work, an approximation for distillation efficiency for an ideal binary can be obtained ... 

The minimum separation work Wmm required for separation is the net change in availability A (A II T()S)... 

The equation (11.13) indicates that a non-ideal mixture with positive deviations (y, > 1) requires less separation work than a mixture with negative deviations (y, <1). For the separation of a mixture in pure components the minimum separation work is ... 

Numerical computations can show that the minimum separation work is rather small. For example, for a binary equimolar mixture the minimum ideal separation work in pure components at 298 K is W i =0.6931 / 7 =1718 J/mol. 

Thermal separation operations are isobaric apart from pressure drops which occur in the separation apparatus. During mixing the change of the free enthalpy AG of the system, the free mixing enthalpy is equal and opposite to the minimum separation work 1V, required to separate the mixture into its pure components. 

AgM is negative for the entire concentration range, as seen in Eq. (1-147). The minimum separation work required to separate 1 mol of the k component mixture is... 

The actual work supplied to separate the mixture is usually higher than the minimum separation work, as given by Eq. (1-148). Additional energy is required to create additional phases if necessary, to divide phases using mechanical means, to mix or to disperse. The energy losses of the separation units, and the pump work necessary to transport the liquid are not considered in... 

The quantity in the square brackets is called the separative duty. The separative duty is a function only of the product, waste, and feed concentrations and amounts. It is a quantitative measure of the separative work done by the cascade. The function (2x — 1) In x/l — x is termed the separation potential. It has a minimum at x = 1/2, where it is equal to zero. At all other values of x the separation potential is positive. The separative duty is always positive. This statement is consistent with the fact that the entropy of isotope mixing is positive. 

Q is expressed in J kg of solid adsorbent. Physically, the grand potential is the free energy change associated with isothermal immersion of fresh adsorbent in the bulk fluid. The absolute value of the grand potential is the minimum isothermal work necessary to clean the adsorbent. Since adsorption occurs spontaneously, the cleaning or regeneration of the adsorbent after it equilibrates with the feed stream is the main operating cost of an adsorptive separation process. 

Most students do not know what is available to solve a separation problem and usually do not know how to do it. This text is intended to be street smart by incorporating all the manipulative techniques practicing chemists use to make the separation work well. Far more separations are described here than can be covered in a one-semester course, but this book is intended to serve as a reference for the remaining separations. The mathematical treatment is kept to a minimum, presenting only what is necessary to illustrate the principles and to show how quantitation and recoveries are determined. Individual instructors will surely supplement the theory based on their interests. 

In the future, it is probable that the supplier of emichment services will permit a customer to specify the assay ( U content) of the tails to which feed is to be stripped so as to minimize the combined cost to the customer of natural UF feed and separative work. F re 12.20 shows qualitatively the effect of tails composition on the contributions to product cost arising from costs for feed and for separative work in stripping and enriching sections. The amount of separative work required in the enriching section is independent of tails composition. But the cost of separative work required in the stripping sections varies from zero when = zp (no stripping) to infinity when xw = 0. Conversely, the cost of feed varies from infinity when Xftf = zp to a minimum at rcn =0, as may be seen from Eq. (12.152). There is therefore an optimum tails assay Xo between Xu = 0 and Xfy = zp, at which the sum of the cost of separative work and the cost of natural uranium feed is a minimum. 

To work out the stage design conditions that lead to minimum unit cost of separative work for a specific barrier type... 

Figure 14.8 shows the dependence of the unit cost of separative work, evaluated from Eq. (14.112), with VjA from (14.113), AjA from (14.115), and QjA from (14.117), on the hi -side pressure p" and the pressure ratio p jp". Minimum unit cost is in the nei borhood of an upstream pressure of 0.55 atm and a pressure ratio of 0.32. There is a considerable range of pressures around this optimum in which the unit cost of separative work changes but little. 

Equation (14.111) for the minimum power of 0.0923 kW to produce 1 kg of separative work per year in uranium isotope separation was derived for cross flow on the low-pressure side of the barrier, with the composition of gas on that side y equal to the composition of the net flow u. The purpose of this section is to show that the minimum power requirement could be reduced further by having v greater than y by an appropriate amount and to derive an expression for the optimum difference between v and y and the corresponding power consumption per unit separative capacity. For this minimum wer case, pressures on the high-pressure and low-pressure sides of the barrier must be so low the only flow through the barrier is of the separating, molecular type, and the mixing efficiency on each side of the barrier is unity. 

Table 14.9 Design conditions and characteristics of gaseous diffusion stage designed for minimum unit cost of separative work ...

The coefficient 4 in Eq. (14.143) for optimum counterflow is to be compared with 5.11 in Eq. (14.110) for cross flow. The minimum possible power input to produce 1 kg of separative work per year is... 

Equations for sqtaiation performance of mass diffusion stage. At the bottom of Fig. 14.35, the stage conditions found to lead to minimum unit cost of separative work are tabulated. The condition that the UP6 content of the two effiuent streams be equal,... 

Hence Ng and Me are equal. Forsberg found that a condition of zero net flow throu the screen led to minimum unit cost of separative work, so that... 

This poor energy utilization compared with gaseous diffusion is inherent in the mass diffusion process. Using a thermodynamic argument similar to Sec. 4.8 for gaseous diffusion, Forsberg showed that the minimum ratio of availability loss rate to rate of production of separative work at any point in a mass diffusion screen is... 

The minimum reversible work requirement of a separation process is solely given by the composition and conditions (T, P) of the feed and product streams [52]. It can be calculated from the difference in Gibbs energy of product and feed... 

An ideal cascade is one which is tapered to yield minimum interstage flow and thus minimum operating cost. In an ideal cascade, the heads stream from the (i — l)th and the tails stream from the (i + 1) th stages are mixed to form the feed to the ith stage. Obviously for best efficiency the two streams should be of identical isotopic composition, otherwise some separative work is... 

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