Fenske-Underwood method

Occasionally there is a need to perform some preliminary but rapid estimates for a specific separation without resorting to the tedious graphical or plate by plate calculations. In such instances one can turn to some of the short-cut methods that have been developed specifically for multicomponent separations in the chemical process industry but which also work reasonably well with binary and multicomponent separations at low temperatures. These are the Fenske-Underwood method for obtaining the minimum number of plates at total reflux, the Underwood method for obtaining the minimum reflux, and the Gilliland correlation to determine the theoretical number of plates based on the information provided by the two prior methods. 

The combined Fenske-Underwood-Gillilland method developed by Frank [100] is shown in Figure 8-47. This relates product purity, actual reflux ratio, and relative volatility (average) for the column to the number of equilibrium stages required. Note that this does not consider tray efficiency, as discussed elsewhere. It is perhaps more convenient for designing new columns than reworking existing columns, and should be used only on at acent-key systems. 

For binary systems or systems that approach binary, the Fenske-Underwood-Erbar/Maddox Method is recommended. For minimum stages, use the Fenske equation.12... 

The minimum reflux ratio, Rm is calculated using Underwood s method (Example 11.16) as 0.83 and, using Fenske s method, Example 11.17, the number of plates at total reflux is nm = 8. The following data have been taken from Figure 11.42, attributable to Gilliland(30) ... 

ESTIMATION OF REFLUX AND NUMBER OF TRAYS (FENSKE-UNDERWOOD-GILLILAND METHOD)... 

The jmethod of O Connell is popular because of its simplicity and the fact that predicted values are conservative (low). It expresses the efficiency in terms of the product of viscosity and relative volatility, pa, for fractionators and the equivalent term HP In for absorbers and strippers. The data on which it is based are shown in Figure 13.43. For convenience of use with computer programs, for instance, for the Underwood-Fenske-Gilliland method which is all in terms in equations not graphs, the data have been replotted and fitted with equations by Ncgahban (University of Kansas, 1985). For fractionators,... 

In the years from 1940 through the 1960s, several notable shortcut fractionation methods were published. Of these, one method that included several of these earlier methods has stood out and is today more accepted. Fenske, Underwood, and Gilliland [9-12] are the core of this proposed method. Yet one more entry is added, the Hengstebeck [13] proposed method to apply multicomponent distillation. As these earlier methods pointed out only two component separations (called binary systems), the Hengstebeck added contribution is most important for multicomponent applications. 

A unique method and result of applying this Fenske-Underwood equation is that, when plotted on log-log graph paper, the components other than key component separations can be determined. Consider Fig. 2.3 notice the straight line drawn through the key component points. The unique thing here is that all other component values are also determined from this straight-line plot. 

For single separation duty, Diwekar et al. (1989) considered the multiperiod optimisation problem and for each individual mixture selected the column size (number of plates) and the optimal amounts of each fraction by maximising a profit function, with a predefined conventional reflux policy. For multicomponent mixtures, both single and multiple product options were considered. The authors used a simple model with the assumptions of equimolal overflow, constant relative volatility and negligible column holdup, then applied an extended shortcut method commonly used for continuous distillation and based on the assumption that the batch distillation column can be considered as a continuous column with changing feed (see Type II model in Chapter 4). In other words, the bottom product of one time step forms the feed of the next time step. The pseudo-continuous distillation model thus obtained was then solved using a modified Fenske-Underwood-Gilliland method (see Type II model in Chapter 4) with no plate-to-plate calculations. The... 

Produce a shortlist of candidates by ranking the alternatives following the total vapor rate. A minimum reflux calculation design based on Fenske-Underwood-Gilliland method should be sufficiently accurate. 

Gilliland (45) used the Fenske method (Sec. 3.2.1) to compute minimum stages, and his own method for computing minimum reflux. However, it was shown (11,48) that the Underwood method (Sec. 3.2.2) for minimum reflux can also be used. 

Overall effect. A procedure for evaluating the overall effect (combining the direct and indirect effects on tray efficiency) was developed by Nelson, Olson, and Sandler (156). This method is based on the Fenske, Underwood, and Gilliland (Eduljee version) shortcut relationships (Secs. 3.2.1 to 3.2.5) and was shown to work well when comparing to a more rigorous procedure. An example (using an x-y diagram) in Sec. 7.3.6 demonstrates how differences between true and apparent volatility affect efficiencies calculated from test data. 

Design a distillation column to separate benzene, toluene, and xylene, using (1) the McCabe-Thiele xy diagram and (2) the Fenske-Underwood-Gilliland (FUG) method. Compare the results with each other. Assume that the system is ideal. 

Compare the results of the McCabe-Thiele and the Fenske-Underwood-Gilliland methods. 

A nomograph for the overall Fenske-Underwood-Gilliland method has been derived that considerably reduces the required calculation effort without undue loss of accuracy. It is based on Fig. 8.5, where the subscripts D and B in the abscissa refer to overhead and bottoms product streams,... 

Equation-Based Design Methods Exact design equations have been developed for mixtures with constant relative volatility. Minimum stages can be computed with the Fenske equation, minimum reflux from the Underwood equation, and the total number of stages in each section of the column from either the Smoker equation (Trans. Am. Inst. Chem. Eng., 34, 165 (1938) the derivation of the equation is shown, and its use is illustrated by Smith, op. cit.), or Underwoods method. A detailed treatment of these approaches is given in Doherty and Malone (op. cit., chap. 3). Equation-based methods have also been developed for nonconstant relative volatility mixtures (including nonideal and azeotropic mixtures) by Julka and Doherty [Chem. Eng. Set., 45,1801 (1990) Chem. Eng. Sci., 48,1367 (1993)], and Fidkowski et al. [AIChE /., 37, 1761 (1991)]. Also see Doherty and Malone (op. cit., chap. 4). 

Table 13-6 shows subsequent calculations using the Underwood minimum reflux equations. The a and Xo values in Table 13-6 are those from the Fenske total reflux calculation. As noted earlier, the % values should be those at minimum reflux. This inconsistency may reduce the accuracy of the Underwood method but to be useful, a shortcut method must be fast, and it has not been shown that a more rigorous estimation of x values results in an overall improvement in accuracy. The calculated firnin is 0.9426. The actual reflux assumed is obtained from the specified maximum top vapor rate of 0.022 kg- mol/s [ 175 lb-(mol/h)] and the calculated D of 49.2 (from the Fenske equation). 

For preliminary studies of batch rectification of multicomponent mixtures, shortcut methods that assume constant molar overflow and negligible vapor and liquid holdup are useful in some cases (see the discussion above concerning the effects of holdup). The method of Diwekar and Madhaven [Ind. Eng. Chem. Res., 30, 713 (1991)] can be used for constant reflux or constant overhead rate. The method of Sundaram and Evans [Ind. Eng. Chem. Res., 32, 511 (1993)] applies only to the case of constant reflux, but is eaty to implement. Both methods employ the Fenske-Underwood-Giluland (FUG) shortcut procedure at successive time steps. Thus, batch rectification is treated as a sequence of continuous, steady-state rectifications. 

The minimum reflux ratio can be evaluated for this two component distillation by using the Fenske-Underwood-Gilliland method and then determining what ratio factor to use to obtain the desired separation using 94 theoretical trays. This approach uses Eq. (15-1), (15-2), (15-3) and (15-4). If this approach is used, Nmm = 21.2 stages and Raun = 2.62. A trial and error calculation with Eq. (15-4) where R is unknown, establishes that a value of 2.75 for R is required to obtain 94 theoretical trays. Thus R = (1.05X2.62) or 2.75 for this colunm. This is reasonable since the ratio ctor for low tonperatures distillation columns is generally between 1.05 and 1.10. 

The simplest distillation models to set up are the shortcut models. These models use the Fenske-Underwood-Gilliland or Winn-Underwood-Gilliland method to determine the minimum reflux and number of stages or to determine the required reflux given a number of trays or the required number of trays for a given reflux ratio. These methods are described in Chapter 11. The shortcut models can also estimate the condenser and reboiler duties and determine the optimum feed tray. 

---