Underwood-Fenske equation

Minimum Trays. This is found with the Fenske-Underwood equation,... 

The Fenske-Underwood equation is first used to calculate the minimum number of theoretical stages [14]. Hdist program code lines 210 and 215 show this equation ... 

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. 

This equation is called the Fenske-Underwood equation. 

A distillation column is designed to separate benzene and toluene. The relative volatility ratio for this binary system is 2.5. If the overhead stream must be at least 97% benzene and the bottoms stream no greater than 6% benzene, apply the Fenske-Underwood equation to estimate the minimum number of stages required. 

The minimum number of trays is found with the Fenske-Underwood equation Nmin = ln [x/(l-x)] fty[x/(l-x)]bf /lna. 

Sun et al. [6] simulated the outlet concentration of each tray, expressed in area-weighted average, versus tray number for a typical run 16552 is shown in Fig. 4.10 and compared with the experimental data. According to the Fenske-Underwood equation under constant relative volatility and low concentration, a plot of logarithmic concentration versus tray number should yield a straight line. In Fig. 4.10,... 

However, the total number of equilibrium stages N, N/N,n, or the external-reflux ratio can be substituted for one of these three specifications. It should be noted that the feed location is automatically specified as the optimum one this is assumed in the Underwood equations. The assumption of saturated reflux is also inherent in the Fenske and Underwood equations. An important limitation on the Underwood equations is the assumption of constant molar overflow. As discussed by Henley and Seader (op. cit.), this assumption can lead to a prediction of the minimum reflux that is considerably lower than the actual value. No such assumption is inherent in the Fenske equation. An exact calculational technique for minimum reflux is given by Tavana and Hansen [Jnd. E/ig. Chem. Process Des. Dev., 18, 154 (1979)]. A computer program for the FUG method is given by Chang [Hydrocarbon Process., 60(8), 79 (1980)]. The method is best applied to mixtures that form ideal or nearly ideal solutions. 

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

To solve Equation 9.51, it is necessary to know the values of not only a ,-j and 9 but also x, d. The values of xitD for each component in the distillate in Equation 9.51 are the values at the minimum reflux and are unknown. Rigorous solution of the Underwood Equations, without assumptions of component distribution, thus requires Equation 9.50 to be solved for (NC — 1) values of 9 lying between the values of atj of the different components. Equation 9.51 is then written (NC -1) times to give a set of equations in which the unknowns are Rmin and (NC -2) values of xi D for the nonkey components. These equations can then be solved simultaneously. In this way, in addition to the calculation of Rmi , the Underwood Equations can also be used to estimate the distribution of nonkey components at minimum reflux conditions from a specification of the key component separation. This is analogous to the use of the Fenske Equation to determine the distribution at total reflux. Although there is often not too much difference between the estimates at total and minimum reflux, the true distribution is more likely to be between the two estimates. 

Another approximation that can be made to simplify the solution of the Underwood Equations is to use the Fenske Equation to approximate xitD. These values of XitD will thus correspond with total reflux rather than minimum reflux. 

Having obtained the minimum number of stages from the Fenske Equation and minimum reflux ratio from the Underwood Equations, the empirical relationship of Gilliland10 can be used to determine the number of stages. The original correlation was presented in graphical form10. Two parameters (X and Y) were used to correlate the data ... 

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). 

In this approach, Fenske s equation [Ind. Eng. Chem., 24, 482 (1932)] is used to calculate which is the number of plates required to make a specified separation at total reflux, i.e., the minimum value of N. Underwood s equations [/. Inst. Pet., 31, 111 (1945) 32,598 (1946) ... 

The Gilliland correlation uses the results of the Fenske and Underwood equations to determine the actual number of equilibrium stages. The correlation has been fit to three equations ... 

The values of and (L/D) - have been previously defined as the minimum number of equilibrium stages (Fenske equation) and minimum reflux ratio (Underwood equation). 

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