Chapter 7. Approximate Shortcut Methods for Multicomponent Distillation

Chapters 5 and 6 served as an introduction to multicomponent distillation. Matrix methods are efficient, but they still require a fair amount of time even on a fast computer. In addition, they are simulation methods and require a known number of stages and a specified feed plate location. Fairly rapid approximate methods are required for preliminary economic estimates, for recycle calculations where the distillation is only a small portion of the entire system, for calculations for control systems, and as a first estimate for more detailed simulation calculations.

In this chapter we first develop the Fenske equation, which allows calculation of multicomponent separation at total reflux. Then we switch to the Underwood equations, which allow us to calculate the minimum reflux ratio. To predict the approximate number of equilibrium stages we then use the empirical Gilliland correlation that relates the actual number of stages to the number of stages at total reflux, the minimum reflux ratio, and the actual reflux ratio. The feed location can also be approximated from an empirical correlation.

1. Derive the Fenske equation, and use it to determine the number of stages required at total reflux and the splits of non-key (NK) components

2. Use the Underwood equations to determine the minimum reflux ratio for multicomponent distillation

3. Use the Gilliland correlation to estimate the actual number of stages in a column and the optimum feed stage location

Equation (7-1) is the definition of the relative volatility applied to the reboiler. Material balances for components A and B around the reboiler are

and

However, at total reflux, B = 0, and L_N = V_R. Thus, the mass balances become

For a binary system this naturally means that the operating line is the y = x line. Combining Eqs. (7-1) and (7-3),

If we now move up the column to stage N, the equilibrium equation is

The mass balances around stage N simplify to y_A,N = x_A,N–1 and y_B,N = x_B,N–1. Combining these equations, we have

Equations (7-4) and (7-5) can be combined to give

which relates the ratio of liquid mole fractions leaving stage N–1 to the ratio in the reboiler.

Repeating this procedure for stage N–1, we obtain

We can alternate between the operating and equilibrium equations until we reach the top stage. The result is

If we define α_AB as the geometric average relative volatility,

Eq. (7-8) becomes

Solving Eq. (7-10) for N_min, we obtain

which is one form of the Fenske equation. N_min is the number of equilibrium contacts including the partial reboiler required at total reflux. If the relative volatility is constant, Eq. (7-11) is exact.

An alternative form of the Fenske equation that is very convenient for multicomponent calculations is easily derived. First, rewrite Eq. (7-11) as

(Dx_A)_dist is equal to the fractional recovery of A in the distillate multiplied by the amount of A in the feed, which is the multicomponent equivalent of Eq. (3-5a), and (Bx_B)_reb is given by the multicomponent equivalent of Eq. (3-5b). Substitution of the multicomponent equivalents of Eqs. (3-5a) to (3-5d) into Eq. (7-12) gives

Equation (7-13) is in a convenient form for multicomponent systems.

The derivation up to this point has been for any number of components. If we now restrict ourselves to a binary system where x_B = 1 – x_A, Eq. (7-11) becomes

where x = x_A is the mole fraction of the more volatile component (MVC). The use of the Fenske equation for binary systems is quite straightforward. With distillate and bottoms mole fractions of the MVC specified, N_min is easily calculated if α_AB is known. If the relative volatility is not constant, α_AB can be estimated from a geometric average as shown in Eq. (7-9). This can be estimated for a first trial as

α_avg = (α_distα_R)^1/2

where α_R is determined from the bottoms composition and α_dist from the distillate composition.

For multicomponent systems calculation with the Fenske equation is straightforward if fractional recoveries of the two keys, A and B, are specified. Equation (7-13) can be used directly to find N_min. The relative volatility can be approximated by a geometric average. Once N_min is known, the fractional recoveries of the NKs can be found by writing Eq. (7-13) for an NK component and either key component. Then solve for (FR_NK,dist) or (FR_NK,bot). When this calculation is done, Eq. (7-13) becomes

Remember that the order of subscripts on α is important.

If two mole fractions are specified, say x_LK,bot and x_HK,dist, the multicomponent calculation is more difficult. We cannot use the Fenske equation directly, but several alternatives are possible. If we can assume that all NKs are nondistributing, we can use the strategy used in Chapter 5 to do mass balances. Assume the NKs follow Eqs. (5-7) to (5-9) and then calculate D and B from the summation equations, Eqs. (3-6a) and (3-6b). Once all distillate and bottoms compositions or values for Dx_i,dist and Bx_i,bot have been found, Eq. (7-11) or (7-12) can be used to find N_min. Use the key components for this calculation. The assumption of nondistribution of the NKs can be checked with Eq. (7-10) or (7-15). If the original assumption is invalid, the calculated value of N_min obtained for key components can be used to calculate the light non-key (LNK) and heavy non-key (HNK) compositions in distillate and bottoms. Then Eq. (7-11) or (7-12) is used again to obtain a more accurate estimate of N_min.

If NKs distribute, a reasonable first guess for the distribution is required. This guess can be obtained by assuming that the distribution of NKs is the same at total reflux as it is at minimum reflux. The distribution at minimum reflux can be obtained from the Underwood equation (Case C) and is covered later.

Accurate use of the Fenske equation obviously requires an accurate value for the relative volatility. Smith (1963) covers in detail a method of calculating α by estimating temperatures and calculating the geometric average relative volatility. Winn (1958) developed a modification of the Fenske equation that allows the relative volatility to vary. For approximate estimates this extra work is seldom necessary.

The development of the Underwood equations is quite complex and is presented in detail by Underwood (1948), Smith (1963), and King (1980). Since for most practicing engineers the details of the development are not as important as the use of the Underwood equations, we will follow the approximate derivation of Thompson (1981). Thus we will outline the important points but wave our hands about the mathematical details of the derivation.

If there are nondistributing HNKs present, a pinch point of constant composition will occur at minimum reflux in the enriching section above where the HNKs are fractionated out. With nondistributing LNKs present, a pinch point will occur in the stripping section. For the enriching section in Figure 7-2, the mass balance for component i is

At the pinch point, where compositions are constant,

The equilibrium expression can be written in terms of K values as

Combining Eqs. (7-16) to (7-18) we obtain a simplified balance valid in the region of constant compositions.

Defining the relative volatility α_i = K_i/K_ref and combining terms in Eq. (7-19),

Solving for the component vapor flow rate, V_Min y_i,j+1, and rearranging

Equation (7-21) can be summed over all components to give the total vapor flow rate in the enriching section at minimum reflux.

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

Since the conditions in the stripping section are different than in the rectifying section, in general and .

Underwood (1948) describes generalized forms of Eqs. (7-22) and (7-23) that are equivalent to defining

Equations (7-22) and (7-23) then become polynomials in ϕ and and have C roots. The equations are now

and

If we assume CMO and constant relative volatilities , Underwood showed there are common values of ϕ and that satisfy both equations. Equations (7-25) and (7-26) can now be added. Thus, at minimum reflux

where α is now an average volatility.

Eq. (7-27) is easily simplified with the external column mass balance

to

ΔV_feed is the change in vapor flow rate at the feed stage. If q is known,

If the feed temperature is specified, a flash calculation on the feed can be used to determine ΔV_feed.

Equation (7-29) is known as the first Underwood equation. It can be used to calculate appropriate values of ϕ. Equation (7-25) is known as the second Underwood equation and is used to calculate V_min. The exact method for using the Underwood equation depends on what can be assumed. Three cases will be considered.

Case A. Assume all NKs do not distribute. In this case the amounts of NKs in the distillate are:

while the amounts of the keys are

Solve Eq. (7-29) for the one value of ϕ between the relative volatilities of the two keys, α_HK–ref < ϕ < α_LK–ref. This value of ϕ can be substituted into Eq. (7-25) to calculate V_min. Then

and L_min is found from mass balance

When there is a sandwich component, there are two ϕ values between α_HK–ref and α_LK–ref, and Case C (discussed later) should be used. The method of Shiras et al. (1950) can be used to check for distribution of NKs.

Case B. Assume that the distributions of NKs determined from the Fenske equation at total reflux are also valid at minimum reflux. In this case the Dx_NK,dist values are obtained from the Fenske equation as described earlier. Again solve Eq. (7-29) for the ϕ value between the relative volatilities of the two keys. This ϕ, the Fenske values of Dx_NK,dist, and the Dx_LK,dist and Dx_HK,dist values from Eqs. (7-31c) and (7-31d) are used in Eq. (7-25) to find V_min. Then Eqs. (7-31e) and (7-32) are used to calculate D and L_min. This procedure is illustrated in Example 7-2. Case C results are probably more accurate.

Case C. Exact solution without further assumptions. Equation (7-29) is a polynomial with C roots. Solve this equation for all values of ϕ lying between the relative volatilities of all components:

α_LNK,1–ref > ϕ₁ > α_LNK,2–ref > ϕ₂ > α_{sandwich–ref} > ϕ₃ > α_LK–ref > ϕ₄ > α_HK–ref > ϕ₅ > α_HNK,1–ref

A sandwich component has a relative volatility between the two keys. This gives C-1 valid roots. Now write Eq. (7-25) C-1 times; once for each value of ϕ. We now have C-1 equations and C-1 unknowns (V_min and Dx_i,dist for all LNKs, sandwich components, and HNKs). Solve these simultaneous equations and then obtain D from Eq. (7-31e) and L_min from Eq. (7-32). A sandwich component problem that must use this approach is given in Problem 7.D15.

In general, Eq. (7-29) is of order C in ϕ where C is the number of components. Saturated liquid and saturated vapor feeds are special cases and, after simplification, are of order C-1. If the resulting equation is quadratic, the quadratic formula can be used to find the roots. Otherwise, a root-finding method or Goal Seek should be employed. If only one root, α_LK–ref > ϕ > α_HK–ref, is desired, a good first guess is to assume ϕ = (α_LK–ref + α_HK–ref)/2. If looking for multiple roots, a good first guess to find the ϕ value between two α values is the average of the two α values.

The results of the Underwood equations will only be accurate if the basic assumption of constant relative volatility and CMO are valid. For small variations in α, a geometric average calculated as

can be used as an approximation. Application of the Underwood equations to systems with multiple feeds was studied by Barnes et al. (1972).

To use the Gilliland correlation we proceed as follows:

1. Calculate N_min from the Fenske equation.

2. Calculate (L/D)_min from the Underwood equations or analytically for a binary system.

3. Choose actual (L/D). This is usually done as some multiplier (1.05 to 1.5) times (L/D)_min.

4. Calculate the abscissa.

5. Determine the ordinate value.

6. Calculate the actual number of stages, N.

The Gilliland correlation should be used only for rough estimates. The calculated number of stages can be off by ±30%, although they are usually within ±7%. Since L/D is usually a multiple of (L/D)_min, L/D = M(L/D)_min, the abscissa can be written as

The abscissa is not very sensitive to the (L/D)_min value but does depend on the multiplier M.

The optimum feed plate location can also be estimated. First, use the Fenske equation to estimate where the feed stage would be at total reflux. This can be done by determining the number of stages required to go from the feed concentrations to the distillate concentrations for the keys.

Now assume that the relative feed location is constant as we change the reflux ratio from total reflux to a finite value. Thus

The actual feed stage can now be estimated from Eq. (7-35b). Since the estimate does not include the effect of feed quality, it is not too accurate.

An alternate procedure that is probably a more accurate estimate of the feed stage location is Kirkbride’s method (Humphrey and Keller, 1997). The ratio of the number of trays above the feed, N_f – 1, to the number below the feed stage, N – N_f, can be estimated as

Since neither procedure includes feed quality in the estimate, they cannot be accurate for all feeds. Best practice is to use these estimates as first guesses of the feed location for simulations.

The Gilliland correlation can also be fit to equations. Liddle (1968) fit the Gilliland correlation to three equations. Variable x is the abscissa value from Eq. (7-34). Then

while for 0.01 < x < 0.90

and for 0.90 ≤ x ≤ 1.0

For most situations Eq. (7-37b) is appropriate. The fit to the data is shown in Figure 7-3. Naturally, the equations are useful for computer calculations. Erbar and Maddox (1961) (see King, 1980, or Hines and Maddox, 1985) developed a somewhat more accurate correlation that uses more than one curve.

As a rough rule of thumb we can estimate N = 2.5 N_min. This estimate then requires only a calculation of N_min and is useful for very preliminary estimates.

Erbar, J. H., and R. N. Maddox, Petrol. Refin., 40 (5), 183 (1961).

Fenske, M. R., “Fractionation of Straight-Run Pennsylvania Gasoline,” Ind. Eng. Chem., 24, 482 (1932).

Gilliland, E. R., “Multicomponent Rectification,” Ind. Eng. Chem., 32, 1220 (1940).

Hines A. L., and R. N. Maddox, Mass Transfer: Fundamentals and Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1985.

Humphrey, J. L., and G. E. Keller II, Separation Process Technology, McGraw-Hill, New York, 1997.

King, C. J., Separation Processes, 2nd ed., McGraw-Hill, New York, 1980.

Liddle, C. J., “Improved Shortcut Method for Distillation Calculations,” Chem. Eng., 75 (23), 137 (Oct. 21, 1968).

Shiras, R. N., D. N. Hansen, and C. H. Gibson, “Calculation of Minimum Reflux in Distillation Columns,” Ind. Eng. Chem., 42, 871 (1950).

Smith, B. D., Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1963.

Thompson, R. E., “Shortcut Design Method-Minimum Reflux,” AIChE Modular Instructions, Series B, Vol. 2, 5 (1981).

Underwood, A. J. V., “Fractional Distillation of Multicomponent Mixtures,” Chem. Eng. Prog., 44, 603 (1948).

Winn, F. W.,“New Relative Volatility Method for Distillation Calculations,” Pet. Refiner, 37, 216 (1958).

A1. The Fenske equation:

a. Is valid only for binary systems.

b. Was derived for minimum reflux.

c. Requires CMO.

d. Requires constant K values.

e. All of the above.

f. None of the above.

A2. If you want to use an average relative volatility, how do you calculate it for the Underwood equation?

A3. Develop your key relations chart for this chapter.

A4. In multicomponent distillation the Fenske equation can be used to:

a. Estimate the fractional recoveries of the NKs at total reflux.

b. Calculate the number of equilibrium contacts at minimum reflux.

c. Estimate the average K value of the LK at total reflux.

d. All of the above.

e. None of the above.

A5. With the ready availability of process simulators, why do chemical engineers still use the Fenske-Underwood-Gilliland (FUG) method?

A6. Suppose you are doing a ternary distillation where component B, the LK, has a 98.3% recovery in the distillate, and component C, the HK, has a 99.8% recovery in the bottoms. If α_A–ref = α_B–ref, how does component A distribute?

C. Derivations

C1. Derive Eq. (7-15). Derive an equation for (FR_C,_bot) in terms of (FR_A,dist).

C2. Derive Eq. (7-30).

C3. If the pinch point occurs at the feed point, mass balances can be used to find the minimum flows. Derive these equations. Note: A pinch point at the feed can occur but is unusual in multicomponent distillation.

C4. The choice of developing the Underwood equations in terms of V_min instead of solving for L_min is arbitrary. Rederive the Underwood equations solving for L_min and . Develop the equations analogous to Eqs. (7-25) and (7-29).

C5. For binary systems, Eq. (7-29) simplifies to a linear equation for both saturated liquid and saturated vapor feeds. Prove this statement.

C6. If NKs do not distribute, you solve the Underwood Eq. (7-29) for ϕ, which satisfies α_LK–ref > ϕ > α_HK–ref. However, if a different reference component is chosen for calculation of the relative volatilities, the value of ϕ changes. Despite the change in ϕ, V_min calculated from Eq. (7-25) is unchanged. The proof that this is true is challenging for the general case but is tractable for a binary system with a saturated liquid feed since Eq. (7-29) becomes linear. Prove for a binary system with a saturated liquid feed that the solution for V_min is not affected by the choice of reference component for relative volatilities.

D. Problems

*Answers to problems with an asterisk are at the back of the book.

D1.* 10.0 kmol/h of a saturated liquid feed that is 40.0 mol% benzene and 60.0 mol% toluene is fed to a column with a partial reboiler and a total condenser. We desire a distillate composition that is 0.992 mole fraction benzene and a bottoms that is 0.986 mole fraction toluene. CMO is valid. Assume constant relative volatility with α_BT = 2.4. Reflux is a saturated liquid.

a. Use the Fenske equation to determine N_min including a partial reboiler.

b. Use the Underwood equations to find (L/D)_min.

c. For L/D = 1.1(L/D)_min, use the Gilliland correlation to estimate the total number of stages and the optimum feed stage location.

D2. A saturated liquid feed at 1000.0 kmol/h that is 8.0 mol% ethane, 33.0 mol% propane, 49.0 mol% n-butane, and 10.0 mol% n-pentane is fed to a distillation column operating at 5.0 atm. The column has a total condenser and a partial reboiler. Recovery of propane in the distillate is 99.7%, and n-butane recovery in the bottoms is 99.8%.

a. Find N_min from the Fenske equation (hand calculation).

b. Find (L/D)_min from the Underwood equation (hand calculation).

c. Use L/D = 1.15(L/D)_min, and estimate N and N_Feed from the Gilliland correlation (hand calculation).

Use DePriester charts. Assume relative volatility is constant at the value calculated at the bubble-point temperature of the feed. For bubble-point calculation, choose propane as the reference component. Choose n-butane (the HK) as the reference component for other calculations.

D3.* A special column acts as exactly three equilibrium stages. Operating at total reflux, we measure vapor composition leaving the top stage and the liquid composition leaving the bottom stage. The column is separating phenol from o-cresol. We measure a phenol liquid mole fraction leaving the bottom stage of 0.36 and a phenol vapor mole fraction leaving the top stage of 0.545. What is the relative volatility of phenol with respect to o-cresol?

D4. Separate 1,2-dichloroethane from 1,1,2-trichloroethane at 1 atm. Distillate is 99.15 mol% 1,2-dichloroethane, and bottoms is 1.773 mol% 1,2-dichloroethane. The saturated liquid feed is 60.0 mol% 1,2-dichloroethane. Relative volatility is approximately constant, α = 2.4.

a. Find the minimum number of stages using the Fenske equation.

b. Calculate L/D_min.

c. Estimate the actual number of stages for L/D = 2.2286 using the Gilliland correlation.

d. A detailed simulation gave 99.15 mol% 1,2-dichloroethane in the distillate, 1.773 mol% 1,2-dichloroethane in the bottoms for L/D = 2.2286, N = 25 equilibrium contacts, and optimum feed location is 16 equilibrium contacts from the top of the column. Compare this N with part c, and calculate the percentage of error in the Gilliland prediction.

D5.* A column with 29 equilibrium stages and a partial reboiler is operated at total reflux to separate ethylene dibromide and propylene dibromide. Ethylene dibromide is more volatile, and the constant relative volatility is 1.30. We determine the distillate concentration is 98.4 mol% ethylene dibromide. The column has a total condenser and a saturated liquid reflux, and CMO can be assumed. Use the Fenske equation to predict the bottoms composition.

D6.* 1000.0 mol/h of a 40.0 mol% benzene and 60.0 mol% toluene is fed into a distillation column with a total condenser and a partial reboiler. The feed is a saturated liquid. CMO is valid. Distillate is 99.3 mol%, benzene and bottoms is 1.0 mol% benzene. Use the Fenske equation to find the number of stages required at total reflux, a McCabe-Thiele diagram to find (L/D)_min, and the Gilliland correlation to estimate the number of stages required if L/D = 1.15(L/D)_min. Relative volatility is constant, α_BT = 2.4. Check your results with a McCabe-Thiele diagram.

D7. Your boss wants some idea of how expensive it will be to distill 155.0 kmol/h of a saturated liquid feed that is 5.0 mol% methane, 10.0 mol% ethane, 15.0 mol% n-butane, 22.0 mol% n-pentane, 22.0 mol% n-hexane, and 26.0 mol% n-heptane. Column pressure is 700.0 kPa. The column has a partial condenser and a partial reboiler. We want to recover 99.0% of the n-butane in the distillate and 98.3% of the n-pentane in the bottoms. Do the calculations of the K values either from the DePriester chart or from Eq. (2-28).

a. Assuming that NKs do not distribute, calculate the values of D and B in kmol/h and the mole fractions in distillate and bottoms.

b. Do a bubble-point calculation at the feed conditions. Calculate the relative volatilities of all components with respect to the HK (n-pentane). Use these values as the average value of relative volatility for the entire column. Also determine the bubble-point temperature of the distillate to see if condensation will be expensive.

c. Determine the minimum number of stages for this separation with the Fenske equation.

d. Determine the minimum reflux ratio, (L/D)_min, with the Underwood method.

e. Estimate the number of stages required if L/D = M × (L/D)_min with the Gilliland correlation (Liddle’s fit is convenient) where M = 1.04, 1.10, and 2.0.

f. Will this distillation be reasonably economical, or should an alternate be found? Briefly explain your reasoning.

Note: Parts b and d are easier to do with a spreadsheet or Wolfram.

D8.* We wish to separate a mixture of 40.0 mol% benzene and 60.0 mol% ethylene dichloride in a distillation column with a partial reboiler and a total condenser. The feed rate is 750 mol/h, and the feed is a saturated vapor. We desire a distillate product of 99.2 mol% benzene and a bottoms product that is 0.5 mol% benzene. Reflux is a saturated liquid, and CMO can be used. Equilibrium data can be approximated with an average relative volatility of 1.11 (benzene is more volatile).

a. Find the minimum external reflux ratio.

b. Use the Fenske equation to find the number of stages required at total reflux.

c. Estimate the total number of stages required for this separation using the Gilliland correlation for L/D = 1.2(L/D)_min.

D9. Separation of propylene from propane is a very important but expensive distillation. Your boss wants to know the effect of changing the column pressure on the number of stages and on the temperature at which the distillate condenses. Use the FUG method to estimate the number of stages required for the following feeds listed. In all cases we want 99.8 mol% propylene in the distillate with a 99.1% recovery of propylene in the distillate. Use either DePriester charts or Eq. (2-28) for K values. Operate with L/D = 1.05(L/D)_min. Report the temperature of the distillate, the minimum number of stages, (L/D)_min, and predicted N_actual. Calculate the average relative volatility, where α = K_propylene / K_propane.

a. Saturated liquid feed, 50 mol% propylene, column pressure is 22.0 bar.

b. Saturated liquid feed, 50 mol% propylene, column pressure is 7.0 bar.

c. Saturated liquid feed, 50 mol% propylene, column pressure is 1.013 bar.

d. Saturated vapor feed, 50 mol% propylene, column pressure is 7.0 bar.

Note: This problem can be solved by brute force, or it can be simplified first and then be easily solved.

D10. We are separating a mixture of benzene, toluene, and cumene at total reflux. We want a 99.9% recovery of cumene in the bottoms and a 99.8% recovery of benzene in the distillate. Predict the recovery of toluene in the bottoms at total reflux. Equilibrium can be represented by constant relative volatilities: α_B-C = 10.71, α_T-C = 4.76,α_C-C = 1.0.

D11. A distillation column with a total condenser and a partial reboiler is separating a mixture of propane (P), n-butane (B), and n-hexane (H). The feed (a saturated vapor) is 20.0 mol% propane, 35.0 mol% n-butane, and 45.0 mol% n-hexane. Feed rate is F = 100.0 kmol/h. We desire a 99.0% recovery of the n-butane in the distillate and a 98.0% recovery of n-hexane in the bottoms. CMO can be assumed valid. If we choose butane as the reference, the average relative volatilities are α_PB = 2.04, α_BB = 1.0, and α_HB = 0.20. Calculate the distillate flow rate D assuming all propane is in distillate, and find the minimum external reflux ratio, (L/D)_min.

D12.* A distillation column with a partial reboiler and a total condenser is being used to separate a mixture of benzene, toluene, and cumene. The feed is 40.0 mol% benzene, 30.0 mol% toluene, and 30.0 mol% cumene. The feed is input as a saturated vapor at 1.0 atm pressure, and the column is at the same pressure. We desire 99.0% recovery of the toluene in the bottoms and 98.0% recovery of the benzene in the distillate. The reflux is returned as a saturated liquid, and CMO can be assumed. Equilibrium can be represented as constant relative volatilities. Choosing toluene (T) as the reference component, α_BT = 2.25 and α_CT = 0.210.

a. Use the Fenske equation to find the number of equilibrium stages required at total reflux and the recovery fraction of cumene in the bottoms.

b. Find the minimum reflux ratio. Use a basis of F = 100.0 kmol/h. State your assumptions.

c. For L/D = 1.25(L/D)_min, use the Gilliland correlation to find the total number of equilibrium stages required. Estimate the optimum feed plate location.

D13.* We have a column separating benzene, toluene, and cumene. The column has a total condenser, a total reboiler, and nine equilibrium stages. The feed is 25.0 mol% benzene, 30.0 mol% toluene, and 45.0 mol% cumene. Feed rate is 100 mol/h, and the feed is a saturated liquid at 1.0 atm. The column pressure is 1.0 atm. The equilibrium data can be represented as constant relative volatilities: α_BT = 2.5, α_TT = 1.0, and α_CT = 0.21. We desire 99.0% recovery of toluene in the distillate and 98.0% recovery of cumene in the bottoms. Determine the required external reflux ratio. If α_BT = 2.25 instead of 2.5, how much will L/D change?

D14. At total reflux a separation requires N_min = 10 equilibrium contacts. At a finite external reflux ratio of L/D = 2.0, the separation requires N = 18 equilibrium contacts. (N and N_min include the partial reboiler and stages in the column but do not include the total condenser.) Find (L/D)_min.

D15.* A distillation column is separating benzene (α = 2.25), toluene (α = 1.00), and cumene (α = 0.21). The column is operating at 101.3 kPa. The column has a total condenser and a partial reboiler, and the optimum feed stage is used. Reflux is a saturated liquid, and L₀/D = 1.2. Feed rate is 1000.0 kmol/h. The saturated liquid feed is 39.7 mol% benzene, 16.7 mol% toluene, and 43.6 mol% cumene. Recover 99.92% of the benzene in the distillate and 99.99% of the cumene in the bottoms. For a first guess to this design problem, use the FUG approach to estimate the optimum feed stage and the total number of equilibrium stages. Note: The Underwood equations must be treated as a Case C problem.

D16.* We are separating a mixture of ethanol and n-propanol. Ethanol is more volatile, and the relative volatility is approximately constant at 2.10. The feed flow rate is 1000.0 kmol/h. The feed is 60 mol% ethanol and is a saturated vapor. We desire x_D = 0.99 mole fraction ethanol and x_B = 0.008 mole fraction ethanol. The reflux is a saturated liquid. There are 30 stages in the column (including the partial reboiler). Use the FUG approach to determine:

a. The number of stages (including partial reboiler) at total reflux.

b. (L/D)_min.

c. (L/D)_actual.

D17. A distillation column is separating toluene and xylene, α = 3.03. The feed is a saturated liquid, and reflux is returned as a saturated liquid. p = 1.0 atm. F = 100.0 kmol/h. Distillate mole fraction is x_D = 0.996, and bottoms x_B = 0.008. Use the Underwood equation to find (L/D)_min and V_min at feed mole fractions of z = 0.1, 0.3, 0.5, 0.7, and 0.9. Check your result at z = 0.5 with a McCabe-Thiele diagram. What are the trends for |Q_c,min| and Q_R,min as the toluene feed concentration increases? Hint: If you write the Underwood equation and solve algebraically for ϕ, the problem is easier than it looks.

D18. A depropanizer has the following feed and constant relative volatilities:

Reflux is a saturated liquid. The feed is a saturated liquid fed at 1.0 kmol/(unit time). Assume CMO.

a.* L/D = 1.5, FR_P,dist = 0.9854, FR_B,bot = 0.8791. Use the FUG method to estimate N.

b. If N = 20, FR_P,dist = 0.9854, and FR_B,bot = 0.8791, estimate the required L/D.

c. For part a find the split of normal hexane at total reflux using N_min.

d. L/D = 1.5, FR_P,dist = 0.999, FR_B,bot = 0.8791. Use the FUG method to estimate N.

Note: Do part a first. Parts of the solution of part a can be reused for the other parts b to d.

D19. A distillation column with a partial reboiler and a total condenser operating at 7.0 bar is separating 100.0 kmol/h of a saturated liquid feed that is 25.0 mol% ethane (C2), 35.0 mol% n-butane (C4), and 40.0 mol% n-pentane (C5). CMO can be assumed valid, and assume that ethane does not distribute. We want 99.2% recovery of n-butane in the distillate and 98.3% recovery of n-pentane in the bottoms. The K values at the distillate are K_C2 = 5.56, K_C4 = 0.655, and K_C5 = 0.234. The K values at the bottoms are K_C2 = 10.67, K_C4 = 2.21, and K_C5 = 0.993. Use the correct average for relative volatilities based on the values of the relative volatilities calculated at distillate and bottoms.

a. Find the distillate mole fractions and the value of the distillate flow rate.

b. Find N_min.

c. Find (L/D)_min.

d. Find N from the Gilliland correlation for M = 1.2.

D20. A distillation column is separating a mixture of benzene, toluene, xylene, and cumene. The feed to the column is 5.0 mol% benzene, 15.0 mol% toluene, 35.0 mol% xylene, and 45.0 mol% cumene. The feed rate is 100.0 kmol/h and is a saturated liquid. We wish to produce a distillate that is 0.57895 mole fraction xylene, 0.07018 mole fraction cumene, and the remainder is toluene and benzene. The bottoms should contain no benzene or toluene. If we select toluene as the reference component, the relative volatilities are approximately constant in the column at the following values: benzene = 2.25, toluene = 1.0, xylene = 0.33, and cumene = 0.21.

a. Find the distillate and bottoms flow rates.

b. Find the number of equilibrium contacts at total reflux.

D21. A distillation column is separating 100.0 kmol/h of a saturated vapor feed that is 30.0 mol% ethanol, 25.0 mol% i-propanol, 35.0 mol% n-propanol, and 10.0 mol% n-butanol at a pressure of 1.0 atm. We want a 98.6% recovery of i-propanol in the distillate and 99.2% recovery of n-propanol in the bottoms. The column has a total condenser and a partial reboiler. For parts b, c, and d, use the FUG method. If we choose n-propanol as the reference, the relative volatilities are ethanol = 2.17, i-propanol = 1.86, n-propanol = 1.0, and n-butanol = 0.412. These relative volatilities can be assumed to be constant.

a. Find D, B, x_i,dist, and x_i,bot.

b. Find N_min and N_F,min.

c. Find (L/D)_min. A spreadsheet is highly recommended to find ϕ.

d. If L/D = 1.10(L/D)_min, find N and the feed stage.

D22. A distillation column with five equilibrium contacts (including the partial reboiler) operating at total reflux is separating benzene (B) and toluene (T) plus an unknown impurity (U). Relative volatilities can be assumed to be constant, and α_B–T = 2.5. The fractional recovery of toluene in the bottoms is FR_T,bot = 0.90. The fractional recovery of unknown in the bottoms is FR_U,bot = 0.95.

a. What is the value of FR_B,dist?

b. What is the value of α_U–T?

D23. A distillation column separates 100.0 kmol/day of a saturated liquid feed that is 20.0 mol% ethanol (E), 35.0 mol% n-propanol (P), and 45.0 mol% n-butanol (But). We want a fractional recovery of butanol in bottoms = 0.972. Bottoms mole fraction butanol x_bot,But = 0.986. Assume relative volatilities are constant: α_E–But = 4.883, α_P–But = 2.336, and α_but–but = 1.0.

a. Determine the flow rates of bottoms, B, and of distillate, D, in kmol/day; and determine the mole fractions of E, P, and But in the bottoms and in the distillate.

b. Find the minimum number of stages, N_min, required for this separation.

c. List any assumption(s) you have made, and justify why they are reasonable. Note that the strongest justification is a calculation, not just words.

D24. A distillation column will separate 100.0 kmol/h of a saturated liquid feed at 200 kPa that is 20.0 mol% propane (Pro), 35.0 mol% n-pentane (Pen), and 45.0 mol% n-hexane (Hex). The column has a total condenser and a partial reboiler. We want a fractional recovery of Hex in the bottoms = 0.983 and a fractional recovery of Pen in distillate of 0.967.

a. Make an appropriate assumption, and determine the flow rates of bottoms, B, and of distillate, D, in kmol/h; and determine the mole fractions of bottoms and of distillate.

b. Determine the bubble-point temperature of the feed, and calculate relative volatilities at this temperature. Use Pen as your reference component. Report the bubble-point temperature, the K values, and the values of relative volatilities. Use a DePriester chart. Show your work.

c. Assume the relative volatilities found in part b are constant, and determine the minimum number of stages, N_min, required for this separation.

d. Do a calculation that justifies why the assumption made in part a is reasonable.

E. More Complex Problems

E1. We are separating 100.0 kmol/h of a saturated liquid feed that is 45.0 mol% propane (P), 15.0 mol% n-butane (B), and 40.0 mol% n-hexane (H). Relative volatilities are α_P–P = 1.0, α_B–P = 0.49, α_H–P = 0.10. At minimum reflux we want fractional recoveries FR_P,dist = 0.995 and FR_H,bot = 0.998. Find (L/D)_min, flow rates D and B, and mole fractions of three components in distillate and bottoms at minimum reflux.

F. Problems Requiring Other Resources

F1. What variables does the Gilliland correlation not include? How might some of these be included? Check the Erbar-Maddox (1961) method (or see King, 1980, or Hines and Maddox, 1985) to see one approach that has been used.

F2. A distillation column with a total condenser and a partial reboiler operates at 1.0 atm.

a. Estimate the number of stages at total reflux to separate nitrogen and oxygen to produce a nitrogen mole fraction in the bottoms of 0.001 and a nitrogen distillate mole fraction of 0.998.

b. If the feed is 79.0 mol% nitrogen and 21.0 mol% oxygen and is a saturated vapor, estimate (L/D)_min.

c. Estimate the number of stages and the feed location if L/D = 1.1(L/D)_min. The column has a total condenser and a partial reboiler.

G. Computer Simulation Problems

G1. Repeat Problem 7.D12 on Aspen Plus using RadFrac and the Peng-Robinson correlation.

a. Find N at total reflux (operate with very small feed and distillate rates and a large L/D).

b. Find (L/D)_min accurately by simulating the process with a few hundred stages.

c. Find the actual number of stages and the optimum feed stage at L/D = 1.25(L/D)_min.

G2. Repeat Problem 7.G1 except use DSTWU in Aspen Plus instead of RadFrac.

Note: At this point in learning Aspen Plus you should be able to pick up some of the simpler new tools on your own. A minor trap in DSTWU is noted at the end of Lab 6.