Minimum reflux ratio, Underwood equation

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

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). 

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 ... 

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

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 ... 

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 ... 

The relative volatilities 0 are defined by Eq. (13-33), is the minimum-reflux ratio L +i/D) ia, and 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 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 ahk and aik, 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 n-C4 and the heavy key is i-Cs. 

Underwood [39] derived Equations 6.27.3 and 6.27.4 for estimating the minimum reflux ratio for a specified separation of two key conponents. These equations assume constant molar overflow and relative volatility. Underwood showed that at minimum reflux the value of 0 in Equations 6.27.3 and 6.27.4 must lie between the relative volatility of the heavy and light key components. If the key components are not adjacent, there will be more than one value of 0. This case is illustrated in an example by Walas [6]. Here, we will assume that the key con5)onents are adjacent. As has been pointed out by Walas [6], the minimum reflux ratio calculated by the Underwood equations could turn out to be negative, which means that the equations do not apply for the given separation. 

Calculate the minimum reflux ratio, from the Underwood equations (Equations 6.27.3 and 6.27.4). 

The next step in the procedure is to calculate the optimum or operating reflux ratio. First, calculate the minimum reflux ratio using the Underwood equations, Equations 6.27.3 and 2.27.4. For the calculation use the geometric average volatility of each component listed in Table 6.27.3. Because flie feed is at its bubble point, q = 1. Thus, Equations 6.27.3 and 6.27.4 becomes... 

The Underwood equation requires a trial-and-error solution and a subsequent material balance to estimate the minimum reflux ratio. First, the unknown, q>, is determined by trial and error, such that both sides of the following equation are equal. The unknown value of q> should lie between the relative volatilities of the light and heavy key components. The key components are those that have their fractional recoveries specified. The most volatile component of the keys is the light key and the least volatile is the heavy key. All other components are referred to as nonkey components. If a nonkey component is lighter than the light key component, it is a light nonkey if it is heavier than the heavy key component it is a heavy nonkey component. 

For multicomponent separations, it is often necessary to estimate the minimum reflux ratio of a fractionating column. A method developed for this purpose by Underwood [10] requires the solution of the equation... 

The equation in its final form is presented here. The reader is referred to the original article (Underwood, 1948) for detailed derivation. The minimum reflux ratio R is given by the equation... 

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. 

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