of possible design configurations. Methods have been developedto calculate both these cases in a rigorous manner.1, 2However, these methods require a computer solution of trayby-tray calculations. In order to begin a detailed design, estimatesof minimum reflux ratio and minimum trays should begenerated from simple methods using a binary key componentanalysis.Minimum StagesThe minimum stages can be calculated for most multicomponentsystems by the Fenske equation.3Sm =log SFlog (aavg)Eq 19-3Sm in this equation includes a partial reboiler and a partialcondenser if these items are used.The aavg is the column average relative volatility of the keycomponents in the separation. Various averaging techniqueshave been proposed such as square root averaging of the a atthe top and bottom of the column. The most common approachis to use an arithmetic average:aavg =atop + abottom2Eq 19-4If volatility varies widely, the approach of Winn4 is suggested,in which a modified volatility is used:ij = KLK / KHK b Eq 19-5where the exponent b is obtained from K-value plots over therange of interest.The minimum stage calculation is:Sm =log??????XDXB???LK???XBXD???HKb???BD???1 - b???log ijEq 19-6Note that Sm includes the partial condenser and partial reboilerif they exist.Minimum Reflux RatioThe Underwood method5 is the most widely used of themethods for calculating minimum reflux ratio. Underwood assumedconstant relative volatility and constant molal overflowin the development of this method. The first step is to evaluate? by trial and error:1 - q = Si = 1nXFi(ai - ?)/aiEq 19-7Once ? is determined, the minimum reflux ratio is:(Lo/D)m + 1 = Rm + 1 = Si = 1nXDi(ai - ?)/aiEq 19-8Number of StagesThe number of theoretical stages required for a given separationat a reflux ratio between minimum and total reflux canbe determined from empirical relationships. Erbar and Maddox6made an extensive investigation of tray-by-tray fractionatorcalculations and developed the correlation inFig. 19-7. This correlation relates the ratio of minimum stagesto theoretical stages to the minimum reflux ratio, Rm, and theoperating reflux ratio, R (where R = Lo/D).Fig. 19-7 can be used to determine an operating reflux for agiven number of stages by entering the figure at the value ofSm/S, moving up to the line representing the value ofRm/(Rm + 1) and reading a value of R/(R + 1). The optimum operatingreflux ratio has been found to be near the minimumreflux ratio. Values of 1.2 to 1.3 times the minimum are common.7 Thus for a given R, a value of S can be determined fromFig. 19-7.This correlation is generated on the basis of bubble pointfeed. If the feed is between the bubble point and dew point thenthe operating reflux should be corrected. Erbar and Maddox6proposed the following relationship to adjust the vapor ratefrom the top tray for non-bubble point feed:Vcorr = Vcalc +???1 -DF???[F(HVF - HBP)]???QCLo???calcEq 19-9The reflux rate can then be adjusted by material balancesince:Lo = V1 - D Eq 19-10Computation MethodIn order to determine the design parameters for a fractionationproblem, the following method is recommended:1. Establish feed composition, flow rate, temperature, andpressure.2. Make product splits for the column and establish condensertemperature and column pressure. From columnpressure, calculate the reboiler temperature.3. Calculate minimum number of theoretical stages fromthe Fenske equation (Eq 19-3).4. Calculate minimum reflux rate from the Underwoodequations (Eq 19-7 and 19-8).5. Obtain theoretical stages/operating reflux relation fromFig. 19-7.6. Adjust actual reflux for feed vaporization if necessary(Eq 19-9, 19-10).Example 19-2 For the given feed stream, 291,000 gal./day(bubble point feed).Desire: 98% C3 in the overhead (relative to the feed)1% iC4 in the overheadAir cooling (120F Condensing Temperature)FeedComposition Mol % Moles/hrC2 2.07 21.5C3 48.67 505.6iC4 10.11 105.0nC4 24.08 250.1iC5 5.41 56.2nC5 4.81 50.0C6 4.85 50.4100.00 1038.8Find the: Minimum trays