Ideal stages

If the feed, solvent, and extract compositions are specified, and the ratio of solvent to feed is gradually reduced, the number of ideal stages required increases. In economic terms, the effect of reducing the solvent-to-feed ratio is to reduce the operating cost, but the capital cost is increased because of the increased number of stages required. At the minimum solvent-to-feed ratio, the number of ideal stages approaches infinity and the specified separation is impossible at any lower solvent-to-feed ratio. In practice the economically optimum solvent-to-feed ratio is usually 1.5 to 2 times the minimum value. 

When the solvents are substantially immiscible and the equiUbrium curve is linear, Y = mX, the number of ideal stages can be calculated without the graphical constmctions (54,55). When the extraction factor S (eq. 1) is not equal to unity, it can be shown that... 

Algebraic Comptttation This method starts with calculation of the quantities and compositions of all the terminal streams, using a convenient quantity of one of the streams as the basis of calculation. Material balance and stream compositions are then computed for a terminal ideal stage at either end of an extraction battery (i.e., at Point A or Point B in Fig. 18-81), using equilibrium and solution-retention data. Calculations are repeated for each successive ideal stage from one end of the system to the other until an ideal stage which corresponds to the desired conditions is obtained. Any solid-hquid extraction problem can be solved by this method. 

For certain simplified cases it is possible to calculate directly the number of stages required to attain a desired product composition for a given set of feed conditions. For example, if equilibrium is attained in all stages and if the underflow mass rate is constant, both the equilibrium and operating lines on a modified McCabe-Thiele diagram are straight, and it is possible to calculate direc tly the number of ideal stages required to accommodate arw rational set of terminal flows and compositions (McCabe, Smith, and Harriott, op. cit.) ... 

Calculate an ideal stage efficiency for a radial and a 45° backward leaning impeller. 

Acetone is to be recovered from an aqueous waste stream by continuous distillation. The feed will contain 10 per cent w/w acetone. Acetone of at least 98 per cent purity is wanted, and the aqueous effluent must not contain more than 50 ppm acetone. The feed will be at 20°C. Estimate the number of ideal stages required. 

For the problem specified in Example 11.2, estimate the number of ideal stages required below an acetone concentration of 0.04 (more volatile component), using the Robinson-Gilliland equation. 

Estimate the number of ideal stages needed in the butane-pentane splitter defined by the compositions given in the table below. The column will operate at a pressure of 8.3 bar, with a reflux ratio of 2.5. The feed is at its boiling point. 

The two most frequently used empirical methods for estimating the stage requirements for multicomponent distillations are the correlations published by Gilliland (1940) and by Erbar and Maddox (1961). These relate the number of ideal stages required for a given separation, at a given reflux ratio, to the number at total reflux (minimum possible) and the minimum reflux ratio (infinite number of stages). 

In each stage calculation it will necessary to estimate the stage temperatures to determine the K values and liquid and vapour enthalpies. The temperature range from top to bottom of the column will be approximately 120 — 60 = 60°C. An approximate calculation (Example 11.7) has shown that around fourteen ideal stages will be needed so the temperature change from stage to stage can be expected to be around 4 to 5°C. 

An estimate of the overall column efficiency will be needed when the design method used gives an estimate of the number of ideal stages required for the separation. 

From Example 11.4, number of ideal stages = 12, one ideal stage will be the reboiler, so number of actual stages... 

Seeds, containing 20 per cent by mass of oil, are extracted in a countercurrent plant and 90 per cent of the oil is recovered as a solution containing 50 per cent by mass of oil. If the seeds are extracted with fresh solvent and 1 kg of solution is removed in the underflow in association with every 2 kg of insoluble matter, how many ideal stages are required ... 

It is then found that xn+ lies between x4 and X5 and hence 4 ideal stages are required. 

By stepping off the ideal stages, the following results are obtained ... 

Determine the number of ideal stages required, and the mass and concentration of the first extract if the final raffinate contains 15 per cent of solute C. 

For a variable underflow the relation between y l and Si, must be determined experimentally as the two curves are no longer straight lines, although the procedure is similar once these have been drawn. Further, it is assumed that each thickener represents an ideal stage and that the ratio of solute to solvent is the same in the overflow and the underflow. If each stage is only 80 per cent efficient, for example, equation 10.30 is no longer applicable, but the same method can be used except that each of the vertical steps will extend only 80 per cent of the way to the curve of y l versus Sh. 

Following the construction described in the text, it is found that point xA coincides exactly with xn+, as shown in Figure 10.23, and hence 3 ideal stages are required. 

It may be seen that partial vaporisation of the liquid gives a vapour richer in the more volatile component than the liquid. If the vapour initially formed, as for instance at point E, is at once removed by condensation, then a liquid of composition x3 is obtained, represented by point C. The step BEC may be regarded as representing an ideal stage, since the liquid passes from composition x2 to a liquid of composition x3, which represents a greater enrichment in the more volatile component than can be obtained by any other single stage of vaporisation. 

In an ideal stage, the extract Ex leaves in equilibrium with the raffinate Rx, so that the point Rx is at the end of the tie line through Ex. To determine the extract E2, PRi is drawn to cut the binodal curve at E2. The points R2, E3, R3, E4, and so on, may be found in the same way. If the final tie line, say ER4, does not pass through R , then the amount of solvent added is incorrect for the desired change in composition. In general, this does not invalidate the method, since it gives the required number of ideal stages with sufficient accuracy. 

Ei along a tie-line and E2 as the projection of PR . The working is continued in this way and it is found that R5 is below R and hence 5 ideal stages are required. 

These units give many ideal stages, run continuously and take up a minimum space. For these reasons they have been adopted in many drug extractions, though they are unsuitable for medium or large throughputs. 

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