Underwoods Method

Commercial Hquid sodium alumiaates are normally analyzed for total alumiaa and for sodium oxide by titration with ethylene diaminetetraacetic acid [60-00-4] (EDTA) or hydrochloric acid. Further analysis iacludes the determiaation of soluble alumiaa, soluble siHca, total iasoluble material, sodium oxide content, and carbon dioxide. Aluminum and sodium can also be determiaed by emission spectroscopy. The total iasoluble material is determiaed by weighing the ignited residue after extraction of the soluble material with sodium hydroxide. The sodium oxide content is determiaed ia a flame photometer by comparison to proper standards. Carbon dioxide is usually determiaed by the amount evolved, as ia the Underwood method. 

The Underwood Method will provide a quick estimate of minimum reflux requirements. It is a good method to use when distillate and bottoms compositions are specified. Although the Underwood Method will be detailed here, other good methods exist such as the Brown-Martin and Colburn methods. These and other methods are discussed and compared in Van Winkle s book. A method to use for column analysis when distillate and bottoms compositions are not specified is discussed by Smith. 

The Underwood Method involves finding a value for a constant, 0, that satisfies the equation... 

The Smith-Brinkley Method uses two sets of separation factors for the top and bottom parts of the column, in contrast to a single relative volatility for the Underwood Method. The Underwood Method requires knowing the distillate and bottoms compositions to determine the required reflux. The Smith-Brinkley Method starts with the column parameters and calculates the product compositions. This is a great advantage in building a model for hand or small computer calculations. Starting with a base case, the Smith-Brinkley Method can be used to calculate the effect of parameter changes on the product compositions. 

Examples chosen for this category include the operations of vapor/liquid separation, heat transfer, and fluid flow. The Underwood method of estimating the... 

Underwood s method (36). This method solves an equation which relates feed composition, thermal condition of the feed, and relative volatility at the average temperature of the column for a factor 6 which lies numerically between the relative volatilities of the keys. This factor is substituted in a second equation which relates minimum reflux to relative volatility and distillate composition. The method assumes constant relative volatility at the mean column temperature and constant molar overflow (Sec. 2.2.2). This method gives reasonable engineering accuracy for systems approaching ideality (28). The Underwood method has traditionally been the most popular for minimum reflux determination, When no distributed key components are present, the method is... 

This compares well with the value of 1.04 obtained from the Hengstebeck diagram (Sec. 2.3.5), and with a value of 1.07 obtained either from the Underwood method, corrected for nonconstant molar overflow (Sec. 3.2.4), or from extrapolation of computer simulation results (Fig. 3.8). 

Graphical Underwood method. To eliminate the trial and error involved, Van Winkle and Todd (37) developed a graphical solution technique for obtaining 8. This technique is only applicable to bubble-point liquid feeds. 

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. 

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). 

Data for the two examples are given in Table I. The i , values, as determined by Eq. (2), appear in Table II. This table shows that the values of R , calculated by Eq. (2) are closest to those of the Underwood method, which is more often the approximation preferred in preliminary multicomponent distillation design. Although it yields slightly lower values, Eq, (2) offers the advantage of being simple and taking less computational time. Further, it does not involve trial and error. 

It is possible to derive—roughly—the equations underlying the Underwood method (Underwood, 1946) from the above. The variable R represents the reflux ratio defined in terms of the liquid flow at the pinch point relative to the distillate top product flow i.e.,... 

The flnal compositions and relative volatilities are then used to calculate the minimum reflux, R, using the Underwood method (Equations 12.29 and 12.30). The calculated values of and are next applied in the Gilliland correlation to determine a suitable combination of trays and reflux, N and R, consistent with economic and design considerations. The column can then be approximately sized based on the internal liquid and vapor flows as calculated from the reflux ratio and product rates. 

The shortcut methods can also be used for approximate analysis of the performance of an existing column. Here, the number of trays, N, is fixed, and the objective is to determine the reflux ratio required to meet a specifled separation. The Fenske and Underwood methods (Equations 12.17, 12.29, and 12.30) are used to calculate the minimum trays and minimum reflux ratio, and R. The operating reflux ratio corresponding to the given number of trays is then read from the Gilliland chart (Figure 12.4). The internal vapor and liquid rates are calculated from the reflux ratio and product rates. A check must be made to determine if the existing column can handle the calculated vapor and liquid traffic. 

This assumption is more restrictive than the assumption of constant relative volatilities, or relative X-values, that is used in the Fenske and Underwood methods. The payback for this assumption is the ability to generalize the model to different degrees of column complexity. The success of the method is dependent on proper evaluation of effective /C-values or other model parameters that would represent actual behavior of the column section. The equilibrium coefficient is commonly lumped with the vapor and liquid molar flows in the column to define the stripping factor. 

For a variable reflux case, the procedure initially assumes constant product (distillate) compositions X, for all the components. The minimum reflux ratio, required to produce a distillate with composition from a feed with composition N+i,i (which is the reboiler initial composition) is calculated by the Underwood method (Equations 12.29 and 12.30). The ratio of refluxes is... 

Given the initial reboiler and the distillate compositions, calculate the minimum reflux ratio and the minimum number of stages by the Fenske and Underwood methods. With a reflux ratio that is twice the minimum, calculate the number of theoretical stages by the Gilliland method. The outcome is the initial reboiler charge and composition for the next step. 

First, the minimum number of stages is computed by the Fenske method. Then, the minimum reflux ratio is computed by the Underwood method. Next, the design (operating) reflux ratio is chosen as some multiple of the minimum reflux ratio, e.g., 1.15 x R. (The optimum multiple is in the... 

The sequences identified by heuristics (Table 7.26) are evaluated by simulation with Aspen Plus using short cut model DSTWU, based on the Underwood method for minimum reflux. In all cases the initial mixture is at 6 bar and 300 K. The pressure in all columns is also at six bar. No intermediate heat exchangers are considered. The thermodynamic option is ideal, based only on vapour pressure data. The measure for total vapour is the reboiler duty. 

Multicomponent mixtures Fenske-Gilliland-Underwood method... 

Determine the minimum reflux by solving the equation (Underwood method) ... 

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