# PROGRAM PROVIDES SHORT-CUT DISTILLATION TOWER CALCULATIONS

A.K. CokerH&G Engineering

Glasgow, U.K.

An interactive Fortran 77 program, named MTRANS, has been developed for short-cut distillation tower calculations based upon the derived method of the Hengstebeck-Geddes equation.

The program has an option of entering either the equilibrium constant or the relative volatility with the feed-stream key component.

It also arranges the relative volatilities in order of decreasing magnitude before proceeding to compute the correlation constants. The following assumptions are used in the program:

- There is only one feed stream with 12 or fewer key components.
- There is only one heavy key component.
- Feed components may be arranged in order of decreasing relative volatility, but the light key (LK) and heavy key (HK) components must be adjacent to each other.
- The feed enters the column at the optimum stage.
- The column produces two products (distillate and bottoms) with overhead condenser and bottom reboiler.

The program uses the bisection method to determine Underwood's constant (Q), or the root of Underwood's equation, and then proceeds to evaluate the minimum reflux ratio. From a given multiplier (FACTOR), the actual reflux ratio is determined.

Fenske's equation determines the minimum number of stages. Gilliland's graphical correlation, as expressed by McCormick in terms of a single equation, computes the actual number of stages.

The location of the feed stage is determined by Kirkbride's empirical equation.

The program can be loaded onto a hard disk of a personal computer, or it can be run from a floppy diskette. The program can be executed directly in the DOS environment.

To run the program from a hard disk system, simply type MTRANS.EXE or MTRANS after the C prompt. If running from a floppy disk, type MTRANS after the A or B prompt.

For results that exceed the normal length of the screen, a facility is provided to allow printing of the results. The program will run only on 80386 microprocessor-based computers (IBM and clones). However, the program listing is on the diskette if users wish to compile the program on other microprocessors.

Presented here are the equations used to develop the program, a comparison of some of the various other equations used to represent Gilliland's graphical correlation, and examples of the program's use.

### EQUATIONS SURVEYED

The increasing availability of personal computers with acquired simulation software packages has helped design engineers to optimize a number of equilibrium stages in multipurpose fractionating towers and absorbers.

Designers adept in mathematical modeling are still engaged in the use of vigorous, iterative plate-to-plate computation to study a wide range of process conditions. However, preliminary design with graphical correlations has often helped to arrive at an approximate optimum number of stages before reverting to established design methods. 12 A schematic of a fractionation column with two or more multicomponent and associated equipment items is as shown in Fig. 1.

Alternatives to the preliminary design are the short-cut methods to achieve a realistic optimum number of theoretical stages. Generally, short-cut methods were proposed to establish the minimum number of stages at total reflux and also the minimum reflux at an infinite number of theoretical stages .3-1

Many authors have correlated the minimum reflux and minimum stages with operating reflux and corresponding theoretical equilibrium stages required. These methods have enabled designers to vary the reflux ratio and plates in order to achieve an optimum relationship based on investment and operating costs.

Of the well-established correlations, Gilliland's has been mostly studied and is used in the program. The correlation relies upon two functions, namely (N - Nm)/(N + 1) and (R - Rm)/(R + 1) respectively.

These functions have been shown to give good correlations independent of the different characteristics of the system involved.

In addition to Gilliland's graphical correlation, the following are also reviewed:

- Estimating the distributions of the non-key components using the Henystebeck-Geddes equation 4
- Determining the minimum number of theoretical equilibrium stages at total reflux
- Employing a numerical method (the bisection method) to establish Underwood's constant, and hence evaluate the minimum reflux ratio
- Determining the location of the feed plate by the Kirkbride equation.

### DERIVED METHOD

The distribution of components between the distillate and bottoms is given by the Hengstebeck-Geddes equation (see Equation 1 in the equation box). A material balance for the ith component in the feed is shown in Equation 2. The quantity of component i in the distillate can be expressed as mole fraction recovered, or di/fi. Alternatively, in the bottoms, the mole fraction of component i recovered is bi/fi.

If Equation 1 is expressed with respect to the heavy key component, it can be expressed as Equation 3. The relative volatility of the heavy key component (aHK) is equal to one, resulting in Equation 4.

But the heavy and light key material balances shown in Equations 5 and 6, result in Equation 7.

Equation 7 can be expressed in terms of mole fraction recovered, as shown in Equation 8.

When Equation 4 is substituted into Equation 1, expressing in terms of the light key component results in Equations 9 and 10. Expressing Equation 10 in terms of fractional recoveries results in Equations 11 and 12.

If Equations 1 and 2 are expressed in terms of the recoveries of the ith component, Equations 13-15 result. Therefore, the recovery of the ith component in the distillate is given by Equation 16, and the recovery of the ith component in the bottoms is given by Equations 17-19.

### FENSKE FOR TOTAL REFLUX

Fenske's equation for determining the minimum equilibrium stages at total reflux was based on an ideal mixture.8 This suggests that the ratio of vapor pressures or the ratio of equilibrium vaporization of the key components is constant over the range of temperatures (the relative volatilities are constant).

Fenske expressed the minimum number of equilibrium stages by the use of Equation 20.

### GILLILAND METHOD

The number of theoretical equilibrium stages required for a given separation at a given reflux ratio is often determined by empirical correlations.' 10 The program uses the Gilliland correlation as shown in Fig. 2.

The abscissa, X represents a reflux function X = (R - Rm)/(R + 1), and the ordinate, Y, represents a stage function Y = (N - Nm)/(N + 1 ).

The ratios used for the axes of abscissa and ordinate were chosen because they provide fixed end points for the curve (X = 0.0, Y 1 .0 and at X = 1.0, Y = 0.0). The two functions were found to give good correlations.

Gilliland's correlation has produced relevant results which offer the following advantages:

- They represent an optimum solution with regard to the location of the feed plate.
- The splitting for the two-key component is verified.
- The maximum deviation using Gilliland's correlation in terms of tray number is within a 7% range.10

### UNDERWOOD'S METHOD

If an infinite, or nearly infinite, number of equilibrium stages is involved, a zone of constant composition must exist in the fractionating column. In this instance, there is no measurable change in composition of liquid or vapor from stage to stage.

Under these conditions, the reflux ratio can be defined as the minimum reflux ratio, Rmin, with respect to a given separation of two-key components (light key and heavy key)." Equation 21 shows Rmin for component i in the distillate. Underwood's constant, 0, or the root of the equation, must lie between the relative volatilities of the light and heavy keys (aHK and (aLK). The number of components is n.

Equation 22 shows the relationship for the feed, where q is the fraction of feed that is liquid at the feed tray temperature and pressure. For a bubble-point feed, q = 1.0, for a dew-point feed, q = 0.

The minimum reflux ratio is determined from Equation 22 by substituting into Equation 21.

### DESCRIBING GILLILAND'S GRAPH

Many equations have been proposed to describe Gilliland's curve for multicomponent distillation. However, the difficulty with some of these equations has been in meeting the end conditions of X 0, Y = 1 and X = 1, Y = 0. A review of the many equations proposed by these authors is shown in Equations 23-34.

From the equations listed, McCormick's gives a good agreement in the normal operating range of real towers. It is, thus, employed in the program.

The reflux ratio, R, is calculated as a multiple of the minimum reflux ratio, Rmin (Equation 35). The multiplier FACTOR generally varies from 1.2 to about 1.5 for conventional columns, but because of economic situations, the range is now between 1.05 to 1.20.

### KIRKBRIDE'S FEED-PLATE LOCATION

After the minimum number of stages and the minimum reflux ratio have been determined, the number of theoretical stages is then calculated. The ratio of the number of plates above the feed stage (including the partial condenser) to the number below the feed stage (including the reboiler) can be obtained using Kirkbride's empirical equation .20

The equation was developed on the basis that the ratio of rectifying trays to stripping trays is a function of:

- The fraction of the heavy-key component (in the feed) removed in the overhead
- The fraction of the light-key component removed in the bottoms
- The concentration of the heavy-key component in the overhead
- The concentration of the light-key component in the bottoms.

Equation 36 shows the relationship.

### EXAMPLES

Three examples are used to show the operation of the program.

Example 1. The feed to a butane-pentane splitter of the composition shown in Table 1 is to be fractionated into a distillate product containing 95% of the n-butane contained in the feed, and a bottoms product containing 95% of the iso-pentane in the feed. The reflux ratio for the fractionation will be 1.3 Rmin and the column pressure 100 psia at the top plate. The reflux and feed are at their bubble-point temperatures. The feed composition and equilibrium constants are shown in Table 1.

The short-cut method is used to determine the recoveries of the components in the distillate and bottoms products, the minimum number of stages, the minimum reflux ratio, the actual number of theoretical plates, and the location of the feed plate. The results of the program execution are shown in Table 2.

Example 2. Propane (LK) and butadiene (HK) are separated in the presence of propylene, butane, and pentane. The desired recovery of both keys is 99% at 400 psia.

The feed is liquid at its bubble-point temperature. If the reflux ratio for the column is 1.37 Rmin, calculate the feed distribution between the column distillate and bottoms, the minimum number of stages, the minimum reflux ratio, the actual number of stages, and the location of the feed plate.

Stream compositions are shown in Table 3, and the results are shown in Table 4.

Example 3. Determine the minimum reflux ratio, minimum number of plates, equilibrium number of plates, and the location of the feed plate for the separation of 3-methylpentene-1, and 2-methylpentene-1 in the presence of isoprene and 2,3-dimethylbutene-2.

Desired recovery of both keys is 99% at 15 psia and the reflux ratio is 1.5 RMIN,. The feed liquid is at its bubble-point temperature.

Stream compositions are shown in Table 5, and results are shown in Table 6.

### ACKNOWLEDGMENTS

The author expresses his appreciation to Mr. E. Sommerville, senior process engineer of H&G Engineering, Glasgow, U.K., for the review of this work, and to H&G Engineering for permission to publish this article.

Editor's note: OGJ subscribers may obtain a free copy of the complete operating program on diskette by sending a blank, 5 1/4 in. floppy diskette, formatted to MS DOS, and a self-addressed, postage-paid or stamped return diskette mailer to: Refining/Petrochemical Editor, Oil & Gas Journal, P.O. Box 1941, Houston, TV U.S.A., 77251.

Subscribers outside of the U.S. send the diskette and return mailer without return postage to the same address. This offer will expire Sept 30, 1990.

### REFERENCES

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