Large Number of Stages

The McCabe-Thiele graphical construction is difficult to apply when conditions of relative volatility and/or product purities are such that a large number of stages must be stepped off. In that event, one of the following techniques can be used to determine the stage requirements. 

Separate plots of expanded scales and/or larger dimensions are used for stepping off stages at the ends of the xy diagram. For example, the additional plots may cover just the regions (1) 0.95 to 1.0, and (2) 0 to 0.05. 

The stages are determined by combining the McCabe-Thiele graphical construction, for a suitable region in the middle, with the Kremser equations for the low and/or high ends, where absorption and stripping factors are almost constant. 

If the equilibrium data are given in analytical form, a McCabe-Thiele computer program can be used. Appendix F is an example of a Mathcad programs to implement the McCabe-Thiele method (Hwalek, 2001). It generates the required VLE data from the Antoine equation for vapor pressure and the NRTL equation for liquid-phase activity coefficients. Appendix F-l is for column feed as saturated liquid Appendix F-2 is for column feed as saturated vapor. 

A methanol (A)-water (B) solution containing 36 mol% methanol is to be continuously rectified at 1 atm pressure at a rate of 216.8 kmol/h (60.22 mol/s) to provide a distillate containing 99.9 mol% methanol and a residue containing 0.5 mol% methanol. The feed to the column is to be preheated to its bubble point. The distillate is to be totally condensed and the reflux returned at the bubble point. A reflux ratio 

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The need for a large number of stages and for the special equipment makes gaseous diffusion an expensive process. The three United States gaseous diffusion plants represent a capital expenditure of close to 2.5 x 10 dollars (17). However, the gaseous diffusion process is one of the more economical processes yet devised for the separation of uranium isotopes on a large scale. 

Where a large number of stages is required, it may be necessary to split a column into two separate columns to reduce the height of the column, even though the required separation could, theoretically, have been obtained in a single column. This may also be done in vacuum distillations, to reduce the column pressure drop and limit the bottom temperatures. 

Conventional distillation tends to be difficult and uneconomical because of the large number of stages required when the relative volatility between the components to be separated is very low. In the extreme case, in which an unwanted azeotrope is formed, distillation past the azeotrope becomes impossible. Extractive or azeotropic distillation can sometimes be used to overcome these difficulties. 

The height of the column section is determined mainly by the number of stages or plates required which in turn depends on how easy or difficult the mixture is to separate, i.e. the extent of separation. If the boiling points of the components to be separated are close, then a large number of stages are required leading to a very tall column. The column height is limited, however, by factors such a wind conditions, etc. 

A more detailed discussion from an economic and operability viewpoint can be found in Doherty and Malone [8]. Despite the apparent advantage, the two-column sequence seems not to be the most economical because of the large entrainer recycle and large number of stages. On the contrary, the two-column sequence plus enrichment column, in total three units, offers the best compromise between investment and solvent-recycle costs. 

Figure 5.5 indicates that the pressure does not affect significantly the composition of azeotropes, although noticeable increase in volatility of cyclohexanone takes place at lower pressure. The y-x diagram of the mixture cyclohexanone/cyclohexa-nol indicates quasi-ideal behavior (Figure 5.6). At atmospheric pressure the relative volatility is so low that the distillation is not feasible. Fortunately, this increases significantly by lowering the pressure. Below 150mm Hg the distillation becomes technically feasible, but still requiring a large number of stages.

The Smoker equation (59) is convenient to use in binary separations with a large number of stages. The equation assumes constant relative volatility and constant molar overflow. The main application is in superfractionators such as ethylene-ethane and isobutane-n-butane separations. The Smoker equation is essentially an analytical solution... 

The mixer-settler unit shown in Fig. 26a is usually considered to be approximately equivalent to one theoretical stage. Some of the typical characteristics of the mixer-settler are shown in Table XIX. For many applications outlined in Table XVIII, a large number of stages (more than five)... 

The technique employed almost exclusively in the process developed is extraction chromatography (LLC), which is an ideal technique, in view of the column exchange capacities, for the extraction of moderate amounts of material present in a limited volume, or for the recovery of elements present in low concentration in large volumes. The simplicity of the equipment and the low sensitivity of extraction performance to fluctuations in solution throughputs make it a simpler technique for use than liquid-liquid extraction. Moreover, LLC lends itself well to discontinuous operation. In the case of difficult separations, such as Am/Cm separation, LLC offers the advantage of a large number of stages in a simple, compact unit. Since the extraction of the U(VI) present in the Masurca waste in macroconcentration is not economically feasible by LLC, the conventional liquid-liquid technique was adopted. 

Make a rough estimate of the number of theoretical stages required. This step employs the procedures developed in Example 8.1. Use the Fenske-Underwood-Gilliland approach rather than the McCabe-Thiele, because the small boiling-point difference indicates that a large number of stages will be needed. 

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