Distillation Underwood equation

Colburn (1941) and Underwood (1948) have derived equations for estimating the minimum reflux ratio for multicomponent distillations. These equations are discussed in Volume 2, Chapter 11. As the Underwood equation is more widely used it is presented in this section. The equation can be stated in the form ... 

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. 

The Underwood Equation is based on the assumption that the relative volatilities and molar overflow are constant between the pinches. Given that the relative volatilities change throughout the column, which are the most appropriate values to use in the Underwood Equations The relative volatilities could be averaged according to Equations 9.47 or 9.49. However, it is generally better to use the ones based on the feed conditions rather than the average values based on the distillate and bottoms compositions. This is because the location of the pinches is often close to the feed. 

Example 11.2 Using the Underwood Equations, determine the best distillation sequence, in terms of overall vapor load, to separate the mixture of alkanes in Table 11.2 into relatively pure products. The recoveries are to be assumed to be 100%. Assume the ratio of actual to minimum reflux ratio to be 1.1 and all columns are fed with a saturated liquid. Neglect pressure drop across each column. Relative volatilities can be calculated from the Peng-Robinson Equation of State with interaction parameters assumed to be zero (see Chapter 4). Determine the rank order of the distillation sequences on the basis of total vapor load for ... 

The errors associated with the Underwood Equations were discussed in Chapter 9, which tend to underpredict the minimum reflux ratio. This introduces uncertainty in the way that the calculations were carried out in Examples 11.2 and 11.3. The differences in the total vapor load between different sequences are small and these differences are smaller than the errors associated with the prediction of minimum reflux ratio and minimum vapor load using the Underwood Equations. However, as long as the errors are consistently low for all of the distillation calculations, the vapor load from the Underwood Equations can still be used to screen between options. Nevertheless, the predictions should be used with caution and options not ruled out because of some small difference in the total vapor load. 

Murphree-stage efficiency Time for distillation Parameter in Underwood equations... 

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. 

Estimate minimum reflux ratio from the Underwood equation, or as an approximation if the distillate is almost pure ... 

Use the Underwood equations to determine the minimum reflux ratio for multicomponent distillation. 

The exact method for using the Underwood equations depends on what can be assumed about the distillation process. Three cases will be considered. 

Use the Underwood equation to estimate the minimum external reflux ratio for the separation by distillation of 30 mole% propane in propylene to obtain 99 mole% propylene and 98mole% propane, if the feed condition at a column operating pressure of 300psia is ... 

Consider the hypothetical perfect separation of a mixture of ethylene and ethane into pure products by distillation as shown below. Two schemes are to be considered conventional distillation and distillation using a heat pump with reboiler liquid flashing. In both cases the column will operate at a pressure of 200 psia, at which the average relative volatility is 1.55. A reflux ratio of 1.10 times minimum, as computed from the Underwood equation, is to be used. Other conditions for the scheme using reboiler liquid flashing are shown below. Calculate for each scheme ... 

For multicomponent mixtures, all components distribute to some extent between distillate and bottoms at total reflux conditions. However, at minimum reflux conditions none or only a few of the nonkey components distribute. Distribution ratios for these two limiting conditions are shown in Fig. 12.14 for the debutanizer example. For total reflux conditions, results from the Fenske equation in Example 12.3 plot as a straight line for the log-log coordinates. For minimum reflux, results from the Underwood equation in Example 12.5 are shown as a dashed line. 

A convenient method for determining the molar vapor rate in an ordinary distillation col umn separating a nearly ideal system uses the Underwood equations to calculate the mini mum reflux ratio, This is readily accomplished, as in the example below, with a proces simulation program. The design reflux ratio is taken as / = 1.2 By material balance, th... 

This value of 0 is substituted into the second Underwood equation, using the distillate composition Xq = 0.98 mol fraction propane. 

D17. A distillation column is separating toluene and xylene, a = 3.03. Feed is a saturated liquid and reflux is returned as a saturated liquid, p = 1.0 atm F = 100.0 kmol/h. Distillate mole fraction is Xp) = 0.996 and bottoms Xg = 0.008. Use the Underwood equation to find (L/D)j and at feed mole fractions of z = 0.1, 0.3, 0.5, 0.7, and 0.9. Gheck your result at z = 0.5 with a McCabe-Thiele diagram What are the trends for Qc,minl Qr nim toluene feed concentration increases ... 

The solution of Underwood gave impetus to numerous investigations based on this approach. Part of these works was directed to the creation of geometric interpretation of the results obtained from the solution of the Underwood equations system. It is impossible without such interpretation to form a true notion of the general regularities of the distillation process of ideal mixtures. For one-section columns, geometric analysis of trajectories, stationary points, and separatrixes of distillation was realized even before the works of Underwood by Hausen (1934, 1935,1952) on the basis of calculations using the method tray by tray. ... 

The task of determining distillation product compositions of ideal mixtures in infinite column at minimum reflux is discussed in the previous section. The Underwood equation system solves this task for set composition xf and thermal state of feeding q at two set parameters (e.g., R and D/F or d, and d,). 

The Underwood equations (Underwood, 1948) provide a shortcut method for determining the minimum reflux ratio, ilmin, in multicomponent distillation under the following assumptions constant relative volatilities and constant molal overflows in the stripping section as well as in the enriching section. The minimum reflux ratio, i min> is obtained from a solution of the following two equations for n components ... 

Underwood equation A shortcut method used to estimate the minimum reflux ratio in a multicomponent distillation process. It was proposed by A. J. V. Underwood in 1948. 

Now the feed, bottoms, and distillate flowrates and compositions are known for both columns. The Fenske equation is used to calculate the minimum number of trays. The actual number of trays is set to 2 times the minimum. The Underwood equations are used to calculate the minimum reflux ratio. For estimating the column diameter, heat exchanger areas, and energy requirements from the vapor rate in the column, the actual reflux ratio is set to 1.2 times the minimum. 

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... 

The Underwood minimum reflux equations of main interest are those that apply when some of the components do not appear in either the distillate or the bottom products at minimum reflux. These equations... 

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