Number of Theoretical Stages

The height of a transfer unit is a function of the type of packing, liquid and gas flow rates, pollutant concentration and solubility, liquid properties, and system temperature. Tower height is primarily a function of packing depth. For most packed-tower applications, e height of a transfer unit can be estimated as 0.3-1.2 m (1-4 ft). The required depth of packing is deter- 

The number of overall transfer units may be estimated graphically by stepping off stages on the equilibrium-operating line graph from inlet conditions to outlet conditions, or by the following equation  

If X = 0 (i.e., a negligible amount of pollutant enters the absorber in the liquid stream) and 1/AF = 0 (i.e., the slope of the equilibrium line is very small and/or the L /G , ratio is very large). Equation 1.26 simplifies to 

There are several methods that may be used to calculate the height of the overall transfer unit all are based on empirically determined packing constants [7-9]. One commonly used method involves determining the overall gas and liquid mass transfer coefficients k, k. A major difficulty in using this approach is that the values of k and /cl is frequently unavailable for specific pollutant-solvent systems of interest. For this purpose, the method used to calculate the height of the overall transfer unit is based on estimating the height of the gas and liquid film transfer units, Hl and Hq, respectively [8-13]. 

---

Fractional vacuum distillation is the method used to separate terpene mixtures into their components. The terpene chemist usually has in the laboratory a range of columns with differing numbers of theoretical stages. Experimental distillation in the laboratory is useful in providing data for manufacturing plants that produce commercial quantities of terpene products. 

McCabe-Thie/e Example. Assume a binary system E—H that has ideal vapor—Hquid equiHbria and a relative volatiHty of 2.0. The feed is 100 mol of = 0.6 the required distillate is x = 0.95, and the bottoms x = 0.05, with the compositions identified and the lighter component E. The feed is at the boiling point. To calculate the minimum reflux ratio, the minimum number of theoretical stages, the operating reflux ratio, and the number of theoretical stages, assume the operating reflux ratio is 1.5 times the minimum reflux ratio and there is no subcooling of the reflux stream, then ... 

Fig. 11. Limiting conditions in binary distillation, (a) Minimum reflux and infinite number of theoretical stages (b) total reflux and minimum number of...

NTU (Number of Transfer Units) The NTU required for a given separation is closely related to the number of theoretical stages or plates required to cariy out the same separation in a stagewise or plate-type apparatus. For equimolal counterdiffusion, such as in a binary distillatiou, the number of overall gas-phase transfer units Nqg required for changing the composition of the vapor stream from yi to yo is... 

The variable that has the most significant impact on the economics of an extractive distillation is the solvent-to-feed (S/F) ratio. For closeboiling or pinched nonazeotropic mixtures, no minimum-solvent flow rate is required to effect the separation, as the separation is always theoretically possible (if not economical) in the absence of the solvent. However, the extent of enhancement of the relative volatihty is largely determined by the solvent concentration and hence the S/F ratio. The relative volatility tends to increase as the S/F ratio increases. Thus, a given separation can be accomplished in fewer equihbrium stages. As an illustration, the total number of theoretical stages required as a function of S/F ratio is plotted in Fig. 13-75 7 for the separation of the nonazeotropic mixture of vinyl acetate and ethyl acetate using phenol as the solvent. 

FIG. 13-75 Number of theoretical stages versus solvent-to-feed ratio for extractive distillation, a) Close-boiling vinyl acetate-etbyl acetate system with phenol solvent, (h) A2eotropic acetone-methanol system with water solvent. 

Computation of Tower Height The required height of a gas-absorption or stripping tower depends on (1) the phase equilibria involved, (2) the specified degree of removal of the solute from the gas, and (3) the mass-transfer efficiency of the apparatus. These same considerations apply both to plate towers and to packed towers. Items 1 and 2 dictate the required number of theoretic stages (plate tower) or transfer units (packed tower). Item 3 is derived from the tray efficiency and spacing (plate tower) or from the height of one transfer unit (packed tower). Solute-removal specifications normally are derived from economic considerations. 

Although Eq. (14-31) is convenient for computing the composition of the exit gas as a function of the number of theoretical stages, an alternative equation derived by Colburn [Tran.s. Am. Jn.st. Chem. Eng., 35, 211 (1939)] is more useful when the number of theoretical plates is the unknown ... 

The main objective for calculating the number of theoretical stages (or mass-transfer units) in the design of a hquid-liquid extraction process is to evaluate the compromise between the size of the equipment, or number of contactors required, and the ratio of extraction solvent to feed flow rates required to achieve the desired transfer of mass from one phase to the other. In any mass-transfer process there can be an infinite number of combinations of flow rates, number of stages, and degrees of solute transfer. The optimum is governed by economic considerations. 

For the system water-acetic acid-MIBK in Fig. 15-11 the raffinate (water) layer is the solubility curve with low concentrations of MIBK, and the extract (MIBK) layer is the solubihty curve with high concentrations of MIBK. The dashed lines are tie lines which connect the two layers in equihbrium as given in Table 15-1. Example 2 describes the right-triangular method of calculating the number of theoretical stages required. 

The end points of the operating line on an XY plot (Fig. 15-13) are X., Y, andXy, Y., and the number of theoretical stages can be stepped off graphically. The equilibrium curve is taken from the Hand type of correlation shown earlier (Fig. 15-9). When the equilibrium line is straight, its intercept is zero, and the operating line is straight, the number of theoretical stages can be calculated with one of the Kremser equations [Eqs. (l5-14) and (15-15)]. When the intercept of the eqnihbrinm line is not zero, the value of YJK, should be used... 

The solution to the Kremser equation is shown graphically in Fig. 15-14. When a system responds with a constant number of theoretic stages N, the solute concentration in the raffinate X can readily be evaluated as the result of changing the ratio of solvent to feed [Eqs. (15-17) and (15-18)]. 

The number of mass-transfer units N r is identical to the number of theoretical stages when the extrac tion fac tor is 1.0 [Eq. (15-24)]. When = 1.0,... 

The overall stage efficiency of a staged extraction system is simply the number of theoretical stages divided by the number of actual stages times 100 [Eq. (15-28)]. 

The height equivalent to a theoretical stage (HETS) in an extraction tower is simply the height of the tower Zt divided by the number of theoretical stages achieved [Eq. (15-29)]. 

Analytical Supercritical Fluid Extraction and Chromatography Supercritical fluids, especially CO9, are used widely to extrac t a wide variety of solid and hquid matrices to obtain samples for analysis. Benefits compared with conventional Soxhlet extraction include minimization of solvent waste, faster extraction, tunabihty of solvent strength, and simple solvent removal with minimal solvent contamination in the sample. Compared with high-performance liquid chromatography, the number of theoretical stages is higher in... 

The number of theoretical stages can then be found in one of the usual ways. Figure 22-43 illustrates a graphical calculation for a stripper. 

Figure 14.1 The McCabe-Thiele diagram for the calculation of the number of theoretical stages required to separate two liquids to yield relatively pure products...

The minimum number of theoretical stages is calculated as follows ... 

The approximate location can be determined by the ratio of the total number of theoretical stages above and below the feed plate from the Fenske total reflux relation ... 

The gas stream shown in Table 8-10 is fed to an isothermal absorber operating at 90°F and 75 psia. 90% of the n-butane is to be removed by contact with a lean oil stream consisting of 98.7 mol% non-volatile oil and the light components shown in Column 2 of Table 8-10. Estimate the composition of the product streams and the required number of theoretical stages if an inlet rate of 1.8 times the minimum is used. 

The stripping factor, Sg, is taken as 1/Ag = 0.6010 4. Calculation of required number of theoretical stages. Using Equation 8-216 for n-butane,... 

The type of distribution to select depends on the sensitivity of the tower performance to the liquid distribution as discussed earlier. Norton s [83] data indicate that the sensitivity of tower performance to liquid distribution quality depends only on the number of theoretical stages in each bed of packing achierable at its System Base HETP [83]. Tower beds of high efficiency packing are more sensitive to liquid distribution quality than shorter beds of medium efficiency packing [83]. It is important to extend the uniformity of the distributor all the way to within one packing particle diameter of the tower wall [85]. 

We may encounter problems in the purification of substances with a high normal boiling point. If purification only requires a small number of theoretical stages. Short Path Distillation (SPD), in which pressures can be as low as 0.001 bar, can prove useful. Many vitamins and pharmaceuticals can be processed without deterioration of quality. It is now common to use mechanical vacuum pumps with proper condensers preceding the pump. 

The actual stage can be a mixing vessel, as in a mixer-settler used for solvent extraction applications, or a plate of a distillation or gas absorption column. In order to allow for non-ideal conditions in which the compositions of the two exit streams do not achieve full equilibrium, an actual number of stages can be related to the number of theoretical stages, via the use of a stage-efficiency factor. 

---