Minimum Reflux Underwood Equations

If there are nondistributing heavy nonkeys present, a pinch point of constant composition will occur at minimum reflux in the rectifying section above where the 

Combining equations (6-69) to (6-71), we obtain a simplified balance valid in the region of constant composition  

Defining the relative volatility or = m/mHK and combining terms in equation (6-72) yields 

Equation (6-74) can be summed over all components to give the total vapor flow in the enriching section at minimum reflux  

In the stripping section, a similar analysis can be used to derive 

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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... 

For an overview approximation and dimensioning, the relationships of Fenske and Underwood can be used to determine the minimum plate number N j and the minimum reflux ratio (Equations 2.3.2-26 and 2.3.2-27) ... 

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. 

The multicomponent form of the Underwood equation can be used to calculate the vapor flow at minimum reflux in each column of the sequence. The minimum vapor rate in a single column is obtained by alternate use of two equations ... 

The relative volatiHties Ot) are defined by Eq. (13-33), is the mini-mum-reflux ratio (L v + i/D)min,. nd q describes the thermal condition of the feed (e.g., 1.0 for a bubble-point feed and 0.0 for a saturated-vapor feed). The Xi p values are available from the given feed composition. The 0 is the common root for the top-section equations and the bottom-section equations developed by Underwood for a column at minimum reflux with separate zones of constant composition in each section. The common root value must fall between 06/, and Ot/, where hk and Ik stand for heavy key and light key respectively. The key components are the ones that the designer wants to separate. In the butane-pentane splitter problem used in Example 1, the light key is /1-C4 and the heavy key is i-C. ... 

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. 

Table 1.3-8 shows subsequent calculations using the Underwood minimum-reflux equations. The (X andxo values in Table 1.3-8 are those from the Fenske... 

Binary minimum reflux so calculated implies feed enthalpy just equal to the above started vapor V and liquid L. Any increase or decrease in that enthalpy must be matehed by inerease or decrease in total heat content of overhead reflux. Note that the Underwood" binary reflux equation essentially computes the flash versus specifi-eation composition relationship along with enthalpy correction. 

Example 8-23 Minimum Reflux Ratio Using Underwood Equation Calculate the Minimiun Reflux Ratio... 

Underwood Algebraic Method, 71 Example 8-23 Minimum Reflux Ratio Using Underwood Equation, 73 Minimum Reflux Colburn Method, 74 Example 8-24 Using the Colburn Equation to Calculate Minimum Reflux Ratio,... 

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 ... 

Minimum reflux ratio Underwood equations 11.60 and 11.61. This calculation is best tabulated. 

For binary or near binary minimum reflux ratio, L/D min, use the Underwood equations.10... 

The Underwood Equations can be used to predict the minimum reflux for multicomponent distillation9. The derivation of the equations is lengthy, and the reader is... 

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. 

The Underwood Equations tend to underestimate the true value of the minimum reflux ratio. The most important reason for this is the assumption of constant molar overflow. As mentioned previously, the Underwood Equations assumed constant molar overflow between the pinches. So far, in order to determine the reflux ratio of the column, this assumption has been extended to the whole column. However, some compensation can be made for the variation in molar overflow by carrying out an energy balance around the top pinch for the column, as shown in Figure 9.16. Thus... 

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 ... 

For the same feed, operating pressure and relative volatility as Exercise 10, the heavy key component is changed to pentane. Now 95% of the propane is recovered in the overheads and 90% of the pentane in the bottoms. Assuming that all lighter than light key components go to the overheads and all the heavier than heavy key go to the bottoms, estimate the distribution of the butane and the minimum reflux ratio using the Underwood Equations. 

In Chapter 9, it was shown how the Underwood Equations can be used to calculate the minimum reflux ratio. A simple mass balance around the top of the column for constant molar overflow, as shown in Figure 11.3, at minimum reflux gives ... 

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. 

The Underwood and Fenske equations may be used to find the minimum number of plates and the minimum reflux ratio for a binary system. For a multicomponent system nm may be found by using the two key components in place of the binary system and the relative volatility between those components in equation 11.56 enables the minimum reflux ratio Rm to be found. Using the feed and top compositions of component A ... 

Then the minimum reflux and the distribution of the intermediate component are found from the two equations that result from substitution of the two values of 8 into Underwood s second... 

Calculate the minimum reflux ratio using the Underwood equations. First solve Eq. (2.106) for the value of the parameter d that lies between aB and ac since the B and C are the light- and heavy-key components in the column. The column feed is assumed to be saturated liquid, so the thermal parameter q is equal to 1 ... 

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... 

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