11.7. DISTILLATION OF MULTICOMPONENT MIXTURES

11.7A. Introduction to Multicomponent Distillation

In industry many of the distillation processes involve the separation of more than two components. The general principles of design for multicomponent distillation towers are the same in many respects as those described for binary systems. There is one mass balance for each component in the multicomponent mixture. Enthalpy or heat balances are made which are similar to those for the binary case. Equilibrium data are used to calculate boiling points and dew points. The concepts of minimum reflux and total reflux as limiting cases are also used.

1. Number of distillation towers needed

In binary distillation, one tower was used to separate the two components A and B into relatively pure components, with A in the overhead and B in the bottoms. However, in a multicomponent mixture of n components, n − 1 fractionators will be required for separation. For example, for a three-component system of components A, B, and C, where A is the most volatile and C the least volatile, two columns will be needed, as shown in Fig. 11.7-1. The feed of A, B, and C is distilled in column 1, and A and B are removed in the overhead and C in the bottoms. Since the separation in this column is between B and C, the bottoms containing C will contain a small amount of B and often a negligible amount of A (often called trace component). The amount of the trace component A in the bottoms can usually be neglected if the relative volatilities are reasonably large. In column 2 the feed of A and B is distilled, with A in the distillate containing a small amount of component B and a much smaller amount of C. The bottoms containing B will also be contaminated with a small amount of C and A. Alternately, column 1 could be used to remove A overhead, with B plus C being fed to column 2 for separation of B and C.

Figure 11.7-1. Separation of a ternary system of A, B, and C.


2. Design calculation methods

In multicomponent distillation, as in binary, ideal stages or trays are assumed in the stage-to-stage calculations. Using equilibrium data, equilibrium calculations are used to obtain the boiling point and equilibrium vapor composition from a given liquid composition or the dew point and liquid composition from a given vapor composition. Material balances and heat balances similar to those described in Section 11.6 are then used to calculate the flows to and from the adjacent stages. These stage-to-stage design calculations involve trial-and-error calculations, and high-speed digital computers are generally used to provide rigorous solutions.

In a design the conditions of the feed are generally known or specified (temperature, pressure, composition, flow rate). Then, in most cases, the calculation procedure follows either of two general methods. In the first method, the desired separation or split between two of the components is specified and the number of theoretical trays is calculated for a selected reflux ratio. It is clear that with more than two components in the feed, the complete compositions of the distillate and bottoms are not then known, and trial-and-error procedures must be used. In the second method, the number of stages in the enriching section and stripping section and the reflux ratio are specified or assumed and the separation of the components is calculated using assumed liquid and vapor flows and temperatures for the first trial. This approach is often preferred for computer calculations (H2, P1). In the trial-and-error procedures, the design method of Thiele and Geddes (P1, S1, T1), which is reliable, is often used to calculate resulting distillate and bottoms compositions together with tray temperatures and compositions. Various combinations and variations of the above rigorous calculation methods are available in the literature (H2, P1, S1) and will not be considered further.

The variables in the design of a distillation column are all interrelated, and there are only a certain number of these which may be fixed in the design. For a more detailed discussion of the specification of these variables, see Kwauk (K2).

3. Shortcut calculation methods

In the remainder of this chapter, shortcut calculation methods for the approximate solution of multicomponent distillation are considered. These methods are quite useful for studying a large number of cases rapidly to help orient the designer, determine approximate optimum conditions, or provide information for a cost estimate. Before discussing these methods, equilibrium relationships and calculation methods of bubble point, dew point, and flash vaporization for multicomponent systems will be covered.

11.7B. Equilibrium Data in Multicomponent Distillation

For multicomponent systems which can be considered ideal, Raoult's law can be used to determine the composition of the vapor in equilibrium with the liquid. For example, for a system composed of four components, A, B, C, and D,

Equation 11.7-1


Equation 11.7-2


In hydrocarbon systems, because of nonidealities, the equilibrium data are often represented by

Equation 11.7-3


where KA is the vapor–liquid equilibrium constant or distribution coefficient for component A. These K values for light hydrocarbon systems (methane to decane) have been determined semiempirically and each K is a function of temperature and pressure. Convenient K factor charts are available from Depriester (D1) and Hadden and Grayson (H1). For light hydrocarbon systems, K is generally assumed not to be a function of composition, which is sufficiently accurate for most engineering calculations. Note that for an ideal system, KA = PA/P, and so on. As an example, data for the hydrocarbons n-butane, n-pentane, n-hexane, and n-heptane are plotted in Fig. 11.7-2 at 405.3 kPa (4.0 atm) absolute (D1, H1).

Figure 11.7-2. Equilibrium K values for light hydrocarbon systems at 405.3 kPa (4.0 atm) absolute.


The relative volatility αi for each individual component in a multicomponent mixture can be defined in a manner similar to that for a binary mixture. If component C in a mixture, of A, B, C, and D is selected as the base component,

Equation 11.7-4


The values of Ki will be a stronger function of temperature than the αi values, since the Ki lines in Fig. 11.7-2 all increase with temperature in a similar manner.

11.7C. Boiling Point, Dew Point, and Flash Distillation

1. Boiling point

At a specified pressure, the boiling point or bubble point of a given multicomponent mixture must satisfy the relation Σ yi = 1.0. For a mixture of A, B, C, and D, with C as the base component,

Equation 11.7-5


The calculation is a trial-and-error process, as follows. First a temperature is assumed and the values of αi are calculated from the values of Ki at this temperature. Then the value of KC is calculated from KC = 1.0/∑ αixi. The temperature corresponding to the calculated value of KC = 1.0/∑ αixi. The temperature corresponding to the calculated value of KC is compared to the assumed temperature. If the values differ, the calculated temperature is used for the next iteration. After the final temperature is known, the vapor composition is calculated from

Equation 11.7-6


2. Dew point

For the dew-point calculation, which is also trial and error,

Equation 11.7-7


The value of KC is calculated from KC = 1.0/∑ αixi. After the final temperature is known, the liquid composition is calculated from

Equation 11.7-8


EXAMPLE 11.7-1. Boiling Point of a Multicomponent Liquid

A liquid feed to a distillation tower at 405.3 kPa abs is fed to a distillation tower. The composition in mole fractions is as follows: n-butane (xA = 0.40), n-pentane (xB = 0.25), n-hexane (xC = 0.20), n-heptane (xD = 0.15). Calculate the boiling point and the vapor in equilibrium with the liquid.

Solution: First a temperature of 65°C is assumed and the values of K obtained from Fig. 11.7-2. Using component C (n-hexane) as the base component, the following values are calculated using Eq. (11.7-5) for the first trial:

Trial 1 (65°C)Trial 3 (70°C) Final
Comp.xiKiαixiKiαiαixiyi
A0.401.686.8572.7431.866.6072.6430.748
B0.250.632.5710.6430.7102.5220.6310.178
C0.200.2451.0000.2000.28151.0000.2000.057
D0.150.0930.3800.0570.1100.3910.0590.017
 1.00 ∑ αixi = 3.643 ∑ αixi = 3.5331.000
KC = 1/∑ αixi = 1/3.643 = 0.2745(69°C)KC = 1/3.533 = 0.2830 (70°C)

The calculated value of KC is 0.2745, which corresponds to 69°C, Fig. 11.7-2. Using 69°C for trial 2, a temperature of 70°C is obtained. Using 70°C for trial 3, the calculations shown give a final calculated value of 70°C, which is the bubble point. Values of yi are calculated from Eq. (11.7-6).


3. Flash distillation of multicomponent mixture

For flash distillation, the process flow diagram is shown in Fig. 11.3-1. Defining f = V/F as the fraction of the feed vaporized and (1 − f) = L/F as the fraction of the feed remaining as liquid, and making a component i balance as in Eq. (11.3-6), the following is obtained:

Equation 11.7-9


where yi is the composition of i in the vapor, in equilibrium with xi in the liquid after vaporization. For equilibrium, yi = Kixi = KCαixi, where αi = Ki/KC. Then Eq. (11.7-9) becomes

Equation 11.7-10


Solving for xi and summing for all components,

Equation 11.7-11


This is solved by trial and error by first assuming a temperature if the fraction f vaporized has been set. When the ∑ xi values add up to 1.0, the proper temperature has been chosen. The composition of the vapor yi can be obtained from yi = KCαixi or by a material balance.

11.7D. Key Components in Multicomponent Distillation

Fractionation of a multicomponent mixture in a distillation tower will allow separation only between two components. For a mixture of A, B, C, D, and so on, a separation in one tower can only be made between A and B, or B and C, and so on. The components separated are called the light key, which is the more volatile (identified by the subscript L), and the heavy key (H). The components more volatile than the light key are called light components and will be present in the bottoms in small amounts. The components less volatile than the heavy key are called heavy components and will be present in the distillate in small amounts. The two key components are present in significant amounts in both the distillate and bottoms.

11.7E. Total Reflux for Multicomponent Distillation

1. Minimum stages for total reflux

Just as in binary distillation, the minimum number of theoretical stages or steps, Nm, can be determined for multicomponent distillation for total reflux. The Fenske equation (11.4-23) also applies to any two components in a multicomponent system. When applied to the heavy key H and the light key L, it becomes

Equation 11.7-12


where xLD is mole fraction of light key in distillate, xLW is mole fraction in bottoms, xHD is mole fraction of heavy key in distillate, and xHW is mole fraction in bottoms. The average value of αL of the light key is calculated from αLD at the top temperature (dew point) of the tower and αLW at the bottoms temperature:

Equation 11.7-13


Note that the distillate dew-point and bottoms boiling-point estimation is partially trial and error, since the distribution of the other components in the distillate and bottoms is not known and can affect these values.

2. Distribution of other components

To determine the distribution or concentration of other components in the distillate and bottoms at total reflux, Eq. (11.7-12) can be rearranged and written for any other component i as follows:

Equation 11.7-14


These concentrations of the other components determined at total reflux can be used as approximations with finite and minimum reflux ratios. More accurate methods for finite and minimum reflux are available elsewhere (H2, S1, V1).

EXAMPLE 11.7-2. Calculation of Top and Bottom Temperatures and Total Reflux

The liquid feed of 100 mol/h at the boiling point given in Example 11.7-1 is fed to a distillation tower at 405.3 kPa and is to be fractionated so that 90% of the n-pentane (B) is recovered in the distillate and 90% of the n-hexane (C) in the bottoms. Calculate the following:

  1. Moles per hour and composition of distillate and bottoms.

  2. Top temperature (dew point) of distillate and boiling point of bottoms.

  3. Minimum stages for total reflux and distribution of other components in the distillate and bottoms.

Solution: For part (a), material balances are made for each component, with component n-pentane (B) being the light key (L) and n-hexane (C) the heavy key (H). For the overall balance,

Equation 11.7-15


For component B, the light key,

Equation 11.7-16


Since 90% of B is in the distillate, yBDD = (0.90)(25) = 22.5. Hence, xBWW = 2.5. For component C, the heavy key,

Equation 11.7-17


Also, 90% of C is in the bottoms, and xCWW = 0.90(20) = 18.0. Then, yCDD = 2.0. For the first trial, it is assumed that no component D (heavier than the heavy key C) is in the distillate and no light A in the bottoms. Hence, moles A in distillate = yADD = 0.40(100) = 40.0. Also, moles D in bottoms = xDWW = 0.15(100) = 15.0. These values are tabulated below.

 Feed. FDistillate, DBottoms, W
Comp.xFxFFyD =xDyDDxWxWW
A0.4040.00.62040.000
B (lt key L)0.2525.00.34922.50.0702.5
C (hy key H)0.2020.00.0312.00.50718.0
D0.1515.0000.423rowsep="1"15.0
 1.00F = 100.01.000D = 64.51.000W = 35.5

For the dew point of the distillate (top temperature) in part (b), a value of 67°C will be estimated for the first trial. The K values are read from Fig. 11.7-2 and the α values calculated. Using Eqs. (11.7-7) and (11.7-8), the following values are calculated:

Comp.yiDKi(67°C)αiyi/αixi
A0.6201.756.730.09210.351
B (L)0.3490.652.500.13960.531
C (H)0.0310.261.000.03100.118
D00.100.38500
 1.000  yii = 0.26271.000
 KC =yii= 0.2627

The calculated value of KC is 0.2627, which corresponds very closely to 67°C, which is the final temperature of the dew point.

For the bubble point of the bottoms, a temperature of 135°C is assumed for trial 1 and Eqs. (11.7-5) and (11.7-6) used for the calculations. A second trial using 132°C gives the final temperature as shown below:

Comp.xiWKiαiαixiyi
A05.004.34800
B (L)0.0702.352.0430.14300.164
C (H)0.5071.151.0000.50700.580
D0.4230.610.5300.22420.256
 1.000 αixi = 0.87421.000
 KC = 1/0.8742 = 1.144

The calculated value of KC is 1.144, which is close to the value at 132°C.

For part (c) the proper α values of the light key L (n-pentane) to use in Eq. (11.7-13) are as follows:

Equation 11.7-13


Substituting into Eq. (11.7-12),


The distribution or compositions of the other components can be calculated using Eq. (11.7-14). For component A, the average α value to use is

Equation 11.7-14


Making an overall balance on A,

Equation 11.7-18


Substituting xADD = 1017xAWW from Eq. (11.7-14) into (11.7-18) and solving,


For the distribution of component D, αD,av = = 0.452.


Solving, xDDD = 0.023, xDWW = 14.977.

The revised distillate and bottoms compositions are as follows:

 Distillate, DBottoms, W
Comp.yD =xDxDDxWxWW
A0.619739.9610.00110.039
B (L)0.348922.5000.07042.500
C (H)0.03102.0000.506818.000
D0.00040.0230.421714.977
 1.0000D = 64.4841.0000W = 35.516

Hence, the number of moles of D in the distillate is quite small, as is the number of moles of A in the bottoms.

Using the new distillate composition, a recalculation of the dew point assuming 67°C gives a calculated value of KC = 0.2637. This is very close to that of 0.2627 obtained when the trace amount of D in the distillate was assumed as zero. Hence, the dew point is 67°C. Repeating the bubble-point calculation for the bottoms assuming 132°C, a calculated value of KC = 1.138, which is close to the value at 132°C. Hence, the bubble point remains at 132°C. If either the bubble- or dew-point temperatures had changed, the new values would then be used in a recalculation of Nm.


11.7F. Shortcut Method for Minimum Reflux Ratio for Multicomponent Distillation

As in the case of binary distillation, the minimum reflux ratio Rm is that reflux ratio which will require an infinite number of trays for the given separation of the key components.

For binary distillation, only one “pinch point” occurs where the number of steps becomes infinite, and that is usually at the feed tray. For multicomponent distillation, two pinch points or zones of constant composition occur: one in the section above the feed plate and another below the feed tray. The rigorous plate-by-plate stepwise procedure for calculating Rm is trial and error and can be extremely tedious for hand calculations.

Underwood's shortcut method for calculating Rm (U1, U2) uses constant average α values and assumes constant flows in both sections of the tower. This method provides a reasonably accurate value. The two equations to be solved in order to determine the minimum reflux ratio are

Equation 11.7-19


Equation 11.7-20


The values of xiD for each component in the distillate in Eq. (11.7-20) are supposed to be the values at the minimum reflex. However, as an approximation, the values obtained using the Fenski total reflux equation are used. Since each αi may vary with temperature, the average value of αi to use in the preceding equations is approximated by using αi at the average temperature of the top and bottom of the tower. Some (P1, S1) have used the average α which is used in the Fenske equation or the α at the entering feed temperature. To solve for Rm, the value of θ in Eq. (11.7-19) is first obtained by trial and error. This value of θ lies between the α value of the light key and the α value of the heavy key, which is 1.0. Using this value of θ in Eq. (11.7-20), the value of Rm is obtained directly. When distributed components appear between the key components, modified methods described by others (S1, T2, V1) can be used.

11.7G. Shortcut Method for Number of Stages at Operating Reflux Ratio

1. Number of stages at operating reflux ratio

The determination of the minimum number of stages for total reflux presented in Section 11.7E and the minimum reflux ratio presented in Section 11.7F are useful for setting the allowable ranges for number of stages and flow conditions. These ranges are helpful in selecting the particular operating conditions for a design calculation. The relatively complex rigorous procedures for doing a stage-by-stage calculation at any operating reflux ratio have been discussed in Section 11.7A.

An important shortcut method for determining the theoretical number of stages required for an operating reflux ratio R is the empirical correlation of Erbar and Maddox (E1) given in Fig. 11.7-3. This correlation is somewhat similar to a correlation by Gilliland (G1) and should be considered as an approximate method. In Fig. 11.7-3 the operating reflux ratio R (for flow rates at the column top) is correlated with the minimum Rm obtained using the Underwood method, the minimum number of stages Nm obtained by the Fenske method, and the number of stages N at the operating R.

Figure 11.7-3. Erbar–Maddox correlation between reflux ratio and number of stages (Rm based on Underwood method). [From J. H. Erbar and R. N. Maddox, Petrol. Refiner, 40(5), 183 (1961). With permission.]


2. Estimate of feed-plate location

Kirkbride (K1) has devised an approximate method to estimate the number of theoretical stages above and below the feed which can be used to estimate the feed-stage location. This empirical relation is as follows:

Equation 11.7-21


where Ne is the number of theoretical stages above the feed plate and Ns the number of theoretical stages below the feed plate.

EXAMPLE 11.7-3. Minimum Reflux Ratio and Number of Stages at Operating Reflux Ratio

Using the conditions and results given in Example 11.7-2, calculate the following:

  1. Minimum reflux ratio using the Underwood method.

  2. Number of theoretical stages at an operating reflux ratio R of 1.5Rm using the Erbar–Maddox correlation.

  3. Location of feed tray using the method of Kirkbride.

Solution: For part (a), the temperature to use for determining the values of αi is the average between the top of 67°C and the bottom of 132°C (from Example 11.7-2) and is (67 + 132)/2, or 99.5°C. The Ki values obtained from Fig. 11.7-2 and the αi values and distillate and feed compositions to use in Eqs. (11.7-19) and (11.7-20) are as follows:

Comp.xiFxiDKi (99.5°C)αi (99.5°C)xiW
A0.400.61973.125.200.0011
B (L)0.250.34891.382.300.0704
C (H)0.200.03100.601.000.5068
D0.150.00040.280.4670.4217
 1.001.0000  1.0000

Substituting into Eq. (11.7-19) with q = 1.0 for feed at the boiling point,

Equation 11.7-22


This is trial and error, so a value of θ = 1.210 will be used for the first trial (θ must be between 2.30 and 1.00). This and other trials are shown below:

 2.080.5750.2000.070 
θ (Assumed)5.2 − θ2.3 − θ1.0 − θ0.467 − θ(Sum)
1.2100.52130.5275−0.9524−0.0942+0.0022
1.2000.52000.5227−1.0000−0.0955−0.0528
1.20960.52130.5273−0.9542−0.0943+0.0001

The final value of θ = 1.2096 is substituted into Eq. (11.7-20) to solve for Rm:


Solving, Rm = 0.395.

For part (b), the following values are calculated: R = 1.5Rm = 1.5(0.395) = 0.593, R/(R + 1) = 0.593/(0.593 + 1.0) = 0.3723, Rm/(Rm + 1) = 0.395/(0.395 + 1.0) = 0.2832. From Fig. 11.7-3, Nm/N = 0.49. Hence, Nm/N = 0.49 = 5.40/N. Solving, N = 11.0 theoretical stages in the tower. This gives 11.0 − 1.0 (reboiler), or 10.0 theoretical trays.

For the location of the feed tray in part (c), using Eq. (11.7-21),

Hence, Ne/Ns = 1.184. Also, Ne + Ns = 1.184Ns + Ns = N = 11.0 stages.

Solving, Ns = 5.0 and Ne = 6.0. This means that the feed tray is 6.0 trays from the top.


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