Chapter 7: Approximate Shortcut Methods for Multicomponent Distillation
The previous chapters 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 will first develop the Fenske equation, which allows calculation of multicomponent separation at total reflux. Then we will 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 the empirical correlation.
7.1 Total Reflux: Fenske Equation
Fenske (1932) derived a rigorous solution for binary and multicomponent distillation at total reflux. The derivation assumes that the stages are equilibrium stages.
Consider the multicomponent distillation column operating at total reflux shown in Figure 7-1, which has a total condenser and a partial reboiler. For an equilibrium partial reboiler for any two components A and B,
Figure7-1 Total reflux column
Equation (7-1) is just the definition of the relative volatility applied to the reboiler. Material balances for these components around the reboiler are
(7-1)
(7-2b)
and
However, at total reflux, B = 0 and LN = VR. Thus the mass balances become
(7-3)
For a binary system this naturally means that the operating line is the y = x line. Combining Eqs. (7-1) and (7-3),
(7-4)
If we now move up the column to stage N, the equilibrium equation is
The mass balances around stage N simplify to
yA,N = xA,N−1 and yB,N = xB,N−1
Combining these equations, we have
(7-5)
Then Eqs. (7-4) and (7-5) can be combined to give
(7-6)
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
(7-7)
We can alternate between the operating and equilibrium equations until we reach the top stage. The result is
(7-8)
If we define αAB as the geometric average relative volatility,
(7-9)
Eq. (7-08) becomes
(7-10)
Solving Eq. (7-10) for Nmin, we obtain
(7-11)
which is one form of the Fenske equation. Nmin 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. Equation (7-11) can also be written as
(7-12)
(DxA)dist is equal to the fractional recovery of A in the distillate multiplied by the amount of A in the feed.
(7-13)
where (FRA)dist is the fractional recovery of A in the distillate. From the definition of fractional recovery,
(7-14)
Substituting Eqs. (7-13) and (7-14) and the corresponding equations for component B into Eq. (7-12) gives
(7-15)
Note that in this form of the Fenske equation, (FRA)dist is the fractional recovery of A in the distillate, while (FRB)bot is the fractional recovery of B in the bottoms. Equation (7-15) 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 xB = 1 − xA, Eq. (7-11) becomes
(7-16)
where x = xA is the mole fraction of the more volatile component. The use of the Fenske equation for binary systems is quite straightforward. With distillate and bottoms mole fractions of the more volatile component specified, Nmin is easily calculated if αAB is known. If the relative volatility is not constant, αAB can be estimated from a geometric average as shown in Eq. (7-9). This can be estimated for a first trial as
αavg = (α1αR)1/2
where αR is determined from the bottoms composition and α1 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 (7-15) can now be used directly to find Nmin. The relative volatility can be approximated by a geometric average. Once Nmin is known, the fractional recoveries of the non-keys (NK) can be found by writing Eq. (7-15) for an NK component, C, and either key component. Then solve for (FRC)dist or (FRC)bot. When this is done, Eq. (7-15) becomes
(7-17)
If two mole fractions are specified, say xLK,bot and xHK,dist, the multicomponent calculation is more difficult. We can’t use the Fenske equation directly, but several alternatives are possible. If we can assume that all NKs are nondistributing, we have
(7-18b)
Equations (7-18) can be solved along with the light key (LK) and heavy key (HK) mass balances and the equations
(7-19)
Once all distillate and bottoms compositions or values for Dxi,dist and Bxi,bot have been found, Eqs. (7-11) or (7-12) can be used to find Nmin. Use the key components for this calculation. The assumption of nondistribution of the NKs can be checked with Eq. (7-10) or (7-17). If the original assumption is invalid, the calculated value of Nmin obtained for key compositions can be used to calculate the LNK and HNK compositions in distillate and bottoms. Then Eq. (7-11) or (7-12) is used again.
If NKs do 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 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. Wankat and Hubert (1979) modified both the Fenske and Winn equations for nonequilibrium stages by including a vaporization efficiency.
EXAMPLE 7:1 Fenske equation
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 mole % benzene, 30 mole % toluene, and 30 mole % cumene and is input as a saturated vapor. We desire 95% recovery of the toluene in the distillate and 95% recovery of the cumene in the bottoms. The reflux is returned as a saturated liquid, and constant molal overflow (CMO) can be assumed. Pressure is 1 atm.
Equilibrium can be represented as constant relative volatilities. Choosing toluene as the reference component, αbenz-tol = 2.25 and αcumene-tol = 0.21. Find the number of equilibrium stages required at total reflux and the recovery fraction of benzene in the distillate.
Solution
A . Define. The problem is sketched below. For A = toluene (LK), B = cumene (HK), C = benzene (LNK), we have αCA = 2.25, αAA = 1.0, αBA = 0.21, zA = 0.3, zB = 0.3, zC = 0.4, FRA,dist = 0.95, and FRB,bot = 0.95.
a . Find N at total reflux.
b . Find FRC,dist at total reflux.
B . Explore. Since operation is at total reflux and relative volatilities are constant, we can use the Fenske equation.
C . Plan. Calculate Nmin from Eq. (7-15), and then calculate FRC,dist from Eq. (7-17).
D . Do It. Equation (7-15) gives
Note that αAB = αtol-cumene = 1/αBA = 1/αcumene-tol. Equation (7-17) gives
which is the desired benzene recovery in the distillate. Note that
E . Check. The results can be checked by calculating FRC,dist using component A instead of B. The same answer is obtained.
F . Generalize. We could continue this problem by calculating Dxi,dist and Bxi,bot for each component from Eqs. (7-13) and (7-14). Then distillate and bottoms flow rates can be found from Eqs. (7-19), and the distillate and bottoms compositions can be calculated.
7.2 Minimum Reflux: Underwood Equations
For binary systems, the pinch point usually occurs at the feed plate. When this occurs, an analytical solution for the limiting flows can be derived (King, 1980) that is also valid for multicomponent systems as long as the pinch point occurs at the feed stage. Unfortunately, for multicomponent systems there will be separate pinch points in both the stripping and enriching sections if there are nondistributing components. In this case an alternative analysis procedure developed by Underwood (1948) is used to find the minimum reflux ratio.
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 (1980). 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
FigureDistillation column
(7-20)
At the pinch point, where compositions are constant,
(7-21)
The equilibrium expression can be written in terms of K values as
(7.22)
Combining Eqs. (7-20) to (7-22) we obtain a simplified balance valid in the region of constant compositions.
(7-23)
Defining the relative volatility αi = Ki/Kref and combining terms in Eq. (7-23),
(7-24)
Solving for the component vapor flow rate, VMin yi,j+1, and rearranging
(7-25)
Equation (7-25) can be summed over all components to give the total vapor flow rate in the enriching section at minimum reflux.
(7-26)
In the stripping section a similar analysis can be used to derive,
(7-27)
Since the conditions in the stripping section are different than in the rectifying section, in general 
and 
.
Underwood (1948) described generalized forms of Eqs. (7-26) and (7-27) which are equivalent to defining
(7-28)
Equations (7-26) and (7-27) then become polynomials in φ and 
and have C roots. The equations are now
(7-29)
and
(7-30)
If we assume CMO and constant relative volatilities 
, Underwood showed there are common values of φ and which satisfy both equations. Equations (7-29) and (7-30) can now be added. Thus, at minimum reflux
where α is now an average volatility.
Eq. (7-31) is easily simplified with the overall column mass balance
(7-32)
to
(7-33)
ΔVfeed is the change in vapor flow rate at the feed stage. If q is known
(7-34)
If the feed temperature is specified a flash calculation on the feed can be used to determine ΔVfeed.
Equation (7-33) is known as the first Underwood equation. It can be used to calculate appropriate values of φ. Equation (7-29) is known as the second Underwood equation and is used to calculate Vmin. Once Vmin is known, Lmin is calculated from the mass balance
(7-35)
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:
DxHNK,dist = 0 and DxLNK,dist = FzLNK
while the amounts of the keys are:
(7-36)
(7-37)
Equation (7-33) can now be solved for the one value of φ between the relative volatilities of the two keys, αHK−ref < φ < αLK−ref. This value of φ can be substituted into Eq. (7-29) to immediately calculate Vmin. Then
(7-38)
And Lmin is found from mass balance Eq. (7-35).
This assumption of nondistributing NKs will probably not be valid for sloppy separations or when a sandwich component is present. In addition, with a sandwich component there are two φ values between αHK and αLK. Thus use Case C (discussed later) for sandwich components. 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 DxNK,dist values are obtained from the Fenske equation as described earlier. Again solve Eq. (7-33) for the φ value between the relative volatilities of the two keys. This φ, the Fenske values of DxNK,dist, and the DxLK,dist and DxHK,dist values obtained from Eqs. (7-36) and (7-37) are used in Eq. (7-29) to find Vmin. Then Eqs. (7-38) and (7-35) are used to calculate D and Lmin. This procedure is illustrated in Example 7-2.
Case C. Exact solution without further assumptions. Equation (7-33) is a polynomial with C roots. Solve this equation for all values of φ lying between the relative volatilities of all components,
αLNK,1 < φ1 < αLNK,2 < φ2 < αLK < φ3 < αHK < φ4 < αHNK,1
This gives C-1 valid roots. Now write Eq. (7-29) C-1 times; once for each value of φ. We now have C-1 equations and C-1 unknowns (Vmin and Dxi,dist for all LNK and HNK). Solve these simultaneous equations and then obtain D from Eq. (7-38) and Lmin from Eq. (7-35). A sandwich component problem that must use this approach is given in Problem 7-D15.
In general, Eq. (7-33) will be 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 (see Problems 7-C7 and 7-C8). If the resulting equation is quadratic, the quadratic formula can be used to find the roots. Otherwise, a root-finding method should be employed. If only one root, αLK > φ > αHK, is desired, a good first guess is to assume φ = (αLK + αHK)/2.
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
(7-39)
can be used as an approximation. Application of the Underwood equations to systems with multiple feeds was studied by Barnes et al. (1972).
EXAMPLE 7:2 Underwood equations
For the distillation problem given in Example 7-1 find the minimum reflux ratio. Use a basis of 100 kg moles/hr of feed.
Solution
A . Define. The problem was sketched in Example 7-1. We now wish to find (L/D)min.
B . Explore. Since the relative volatilities are approximately constant, the Underwood equations can easily be used to estimate the minimum reflux ratio.
C . Plan. This problem fits into Case A or Case B. We can calculate Dxi,dist values as described in Cases A or B, Eqs. (7-36) and (7-37), and solve Eq. (7-33) for φ where φ lies between the relative volatilities of the two keys 0.21 < φ < 1.00. Then Vmin can be found from Eq. (7-29), D from Eq. (7-38) and Lmin from Eq. (7-35).
Do It. Follow Case B analysis. Since the feed is a saturated vapor, q = 0 and ΔVfeed = F (1 − q) = F = 100 and Eq. (7-33) becomes
Solving for φ between 0.21 and 1.00, we obtain φ = 0.5454. Equation (7-29) is
where
Dxi,dist = F zi(FR)i,dist
For benzene this is
Dxben,dist = 100(0.4)(0.998) = 39.92
where the fractional recovery of benzene is the value calculated in Example 7-1 at total reflux. The other distillate values are
Dxtol,dist = 100(0.3)(0.95) = 28.5 and Dxcum,dist = 100(0.3)(0.05) = 1.5
Summing the three distillate flows, D = 69.92. Equation (7-29) becomes
From a mass balance, Lmin = Vmin − D = 44.48, and (L/D)min = 0.636.
E . Check. The Case A calculation gives essentially the same result.
F . Generalize. The addition of more components does not make the calculation more difficult as long as the fractional recoveries can be accurately estimated. The value of φ must be accurately determined since it can have a major effect on the calculation. Since the separation is easy, (L/D)min is quite small in this case. (L/D)min will not be as dependent on the exact values of φ as it is when (L/D)min is large.
7.3 Gilliland Correlation for Number of Stages at Finite Reflux Ratio
A general shortcut method for determining the number of stages required for a multicomponent distillation at finite reflux ratios would be extremely useful. Unfortunately, such a method has not been developed. However, Gilliland (1940) noted that he could empirically relate the number of stages N at finite reflux ratio L/D to the minimum number of stages Nmin and the minimum reflux ratio (L/D)min. Gilliland did a series of accurate stage-by-stage calculations and found that he could correlate the function (N−Nmin)/(N + 1) with the function [L/D − (L/D)min]/(L/D + 1). This correlation as modified by Liddle (1968) is shown in Figure 7-3. The data points are the results of Gilliland’s stage-by-stage calculations and show the scatter inherent in this correlation.
Figure7-3 Gilliland correlation as modified by Liddle (1968); reprinted with permission from Chemical Engineering, 75(23), 137 (1968), copyright 1968, McGraw-Hill.
To use the Gilliland correlation we proceed as follows:
1. Calculate Nmin 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 only be used 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.
(7-40a)
Now assume that the relative feed location is constant as we change the reflux ratio from total reflux to a finite value. Thus
(7-40b)
The actual feed stage can now be estimated from Eq. (7-40b).
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, Nf – 1, to the number below the feed stage, N – Nf, can be estimated as,
(7-41)
Since neither procedure is likely to be very accurate, they should only be used 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. Let x = [L/D − (L/D)min]/(L/D + 1). Then
(7-42a)
while for 0.01 < x < 0.90
(7-42b)
and for 0.90 ≤ x ≤ 1.0
(7-42c)
For most situations Eq. (7-42b) is appropriate. The fit to the data is shown in Figure 7-3. Naturally, the equations are useful for computer calculations.
As a rough rule of thumb we can estimate N = 2.5 Nmin. This estimate then requires only a calculation of Nmin and will be useful for very preliminary estimates. Erbar and Maddox (1961) (see King, 1980 or Hines and Maddox, 1985) developed a somewhat more accurate correlation which uses more than one curve.
EXAMPLE 7:3 Gilliland correlation
Estimate the total number of equilibrium stages and the optimum feed plate location required for the distillation problem presented in Examples 7-1 and 7-2 if the actual reflux ratio is set at L/D = 2.
Solution
A . Define. The problem was sketched in Examples 7-1 and 7-2. F = 100, L/D = 2, and we wish to estimate N and NF.
B . Explore. An estimate can be obtained from the Gilliland correlation, while a more exact calculation could be done with a process simulator. We will use the Gilliland correlation.
C . Plan. Calculate the abscissa
from the Gilliland correlation, and then find N. (L/D)min was found in Example 7-2, and Nmin in Example 7-1. The feed plate location is estimated from Eqs. (7-41) and (7-40).
Do It.
The corresponding ordinate (N − Nmin)/(N + 1) = 0.27 using Liddle’s curve. Since Nmin = 3.77, N = 5.53. From Eq. (7-40a), NF,min is calculated as
Where xLK,dist was found from Example 7-2 as
and
xHK,dist = xcum,dist = 0.021
Then, from Eq. (7-40b),
E . Check. A check of the Gilliland correlation can be obtained from Eq. (7-42b). With x = 0.455 this is
or (1 − 0.283) N = Nmin + 0.283, which gives N = 5.65. The 2% difference between these two results gives an idea of the accuracy of Eq. (7-42) in fitting the curve.
A check on the value of Nf can be obtained with Kirkbride’s Eq. (7-41). To use this equation we need to know the terms on the RHS. From Example 7-2, F = 100 and D = 69.92. Thus, B = 100 − 69.92 = 30.08. The HK = cumene and the LK = toluene. The feed mole fractions of both are 0.30. From Example 7-2:
Dxcum,dist = 1.5. Then xHK,dist = (Dxcum,dist)/D = 1.5/69.92 = 0.02145.
Dxtol,dist = 28.5. Then Bxtol,bot = Fz − Dxtol,dist = 30.0 − 28.5 = 1.5, and
xLK,bot = Bxtol,bot /B = 1.5/30.08 = 0.04987.
Then, Eq. (7-41) becomes,
and (Nf − 1)(N − Nf) = 1.1898
which gives Nf = 3.46 if we use N = 5.53 or Nf = 3.7 if we use N = 6. Thus, the best estimate is to use either the 3rd or 4th stage for the feed. This agrees rather well with the previous estimate.
A complete check would require solution with a process simulator.
F . Generalize. The Gilliland correlation is a rapid method for estimating the number of equilibrium stages in a distillation column. It should not be used for final designs because of its inherent inaccuracy.
7.4 Summary – Objectives
In this chapter we have developed approximate shortcut methods for binary and multicomponent distillation. You should be able to satisfy the following objectives:
1. Derive the Fenske equation and use it to determine the number of stages required at total reflux and the splits of 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
References
Barnes, F.J., D.N. Hansen, and C.J. King, “Calculation of Minimum Reflux for Distillation Columns with Multiple Feeds,” Ind. Eng. Chem. Process Des. Develop., 11, 136 (1972).
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).
Wankat, P.C. and J. Hubert, “Use of the Vaporization Efficiency in Closed Form Solutions for Separation Columns,” Ind. Eng. Chem. Process Des. Develop., 18, 394 (1979).
Winn, F.W., Pet. Refiner, 37, 216 (1958).
Homework
A Discussion Problems
A1. What assumptions were made to derive Fenske Eqs. (7-12) and (7-15)?
A2. If you want to use an average relative volatility how do you calculate it for the Fenske equation? For the Underwood equation?
A3. Develop your key relations chart for this chapter.
C . Derivations
C1. Derive Eq. (7-15).
C2. Derive Eq. (7-17). Derive an equation for (FRC)bot in terms of (FRA)dist.
C3. Derive Eq. (7-34).
C4. Check the accuracy of Eq. (7-42).
C5. If the pinch point occurs at the feed point, mass balances can be used to find the minimum flows. Derive these equations.
C6. The choice of developing the Underwood equations in terms of Vmin instead of solving for Lmin is arbitrary. Rederive the Underwood equations solving for Lmin and Lmin. Develop the equations analogous to Eqs. (7-29) and (7-33).
C7. For binary systems, Eq. (7-33) simplifies to a linear equation for both saturated liquid and saturated vapor feeds. Prove this.
C8. For ternary systems, Eq. (7-33) simplifies to a quadratic equation for both saturated liquid and saturated vapor feeds. Prove this.
D . Problems
*Answers to problems with an asterisk are at the back of the book.
D1.* We have 10 kg moles/hr of a saturated liquid feed that is 40 mole % benzene and 60 mole % toluene. We desire a distillate composition that is 0.992 mole fraction benzene and a bottoms that is 0.986 mole fraction toluene (note units). CMO is valid. Assume constant relative volatility with αBT = 2.4. Reflux is returned as a saturated liquid. The column has a partial reboiler and a total condenser.
a . Use the Fenske equation to determine Nmin.
b . Use the Underwood equations to find (L/D)min.
c . . For L/D = 1.1(L/D)min, use the previous results and the Gilliland correlation to estimate the total number of stages and the optimum feed stage location.
D2. Use the Fenske-Underwood-Gilliland (FUG) method to design a column for the system in problem 5.D6.
a . Do problem 5.D6 and determine external balances and calculate distillate and bottoms mole fractions and flow rates.
b . Determine the temperatures of the top stage and of the partial reboiler. The top stage will be a dew point calculation and the bottom stage will be a bubble point calculation.
c . At both the top and the reboiler calculate the relative volatilities of methanol to butanol and of ethanol to butanol. Calculate the geometric average relative volatilities.
c . Use the Fenske equation to obtain the stages required at total reflux.
d . Use the Underwood equation to obtain the minimum reflux ratio.
e . Estimate the number of stages at the actual reflux ratio with the Gilliland correlation.
D3.* We have designed a special column that 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. We desire to separate 1,2 dichloroethane from 1,1,2 trichloroethane at one atmosphere. The feed is a saturated liquid and is 60.0 mole % 1,2 dichloroethane. Assume the relative volatility is approximately constant, α = 2.4.
a . Find the minimum number of stages using the Fenske equation.
b . Calculate L/Dmin.
c . Estimate the actual number of stages for L/D = 2.2286 using the Gilliland correlation.
d . A detailed simulation gave 99.15 mole % dichloroethane in the distillate and 1.773% dichloroethane in the bottoms for L/D = 2.2286, N = 25 equilibrium contacts, optimum feed location is 16 equilibrium contacts from the top of the column. Compare this N with part c and calculate the % error in the Gilliland prediction. Hypothesize why the error is so large.
D5.∗ A column with 29 equilibrium stages and a partial reboiler is being operated at total reflux to separate a mixture of ethylene dibromide and propylene dibromide. Ethylene dibromide is more volatile, and the relative volatility is constant at a value of 1.30. We are measuring a distillate concentration that is 98.4 mole % ethylene dibromide. The column has a total condenser and saturated liquid reflux, and CMO can be assumed. Use the Fenske equation to predict the bottoms composition.
D6.* We are separating 1,000 moles/hr of a 40% benzene, 60% toluene feed in a distillation column with a total condenser and a partial reboiler. Feed is a saturated liquid. CMO is valid. A distillate that is 99.3% benzene and a bottoms that is 1% benzene are desired. 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. Estimate that the relative volatility is constant at αBT = 2.4. Check your results with a McCabe-Thiele diagram.
D7. We are separating a mixture of ethane, propane, n-butane and n-pentane in a distillation column operating at 5.0 atm. The column has a total condenser and a partial reboiler. The feed flow rate is 1000.0 kmoles/hr. The feed is a saturated liquid. Feed is 8.0 mole % ethane, 33.0 mole % propane, 49.0 mole % n-butane and 10.0 mole % n-pentane. A 98.0% recovery of propane is desired in the distillate. A 99.2% recovery of n-butane is desired in the bottoms. Use the optimum feed stage. Use DePriester chart. Assume relative volatility is constant at value calculated at the bubble point temperature of the feed.
a . Find Nmin from the Fenske equation
b . Find (L/D)min from the Underwood equation
c . Use L/D = 1.2 (L/D)min and estimate N and NFeed from Gilliland correlation.
D8.* We wish to separate a mixture of 40 mole % benzene and 60 mole % ethylene dichloride in a distillation column with a partial reboiler and a total condenser. The feed rate is 750 moles/hr, and feed is a saturated vapor. We desire a distillate product of 99.2 mole % benzene and a bottoms product that is 0.5 mole % 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. We are separating a mixture of ethanol and water in a distillation column with a total condenser and a partial reboiler. Column is at 1.0 atm. pressure. The feed is 30.0 mole % ethanol. The feed is a two-phase mixture that is 80 % liquid. Feed rate is 100.0 kmoles/hr. We desire a bottoms concentration of 2.0 mole % ethanol, and a distillate that is 80.0 mole % ethanol.
Find a . (L/D)min
b . Nmin
c . If (L/D) = 1.05 (L/D)min estimate N and the feed location.
Do this problem by using McCabe-Thiele diagram for parts a and b and the Gilliland correlation for part c. Equilibrium data are given in Table 2-1.
D10. A distillation column is separating ethane (E), propane (P) and n-butane (B) at 5 atm. Operation is at total reflux. We want a 99.2% recovery of ethane in the distillate and a 99.5% recovery of n-butane in the bottoms. Propane is a sandwich component (e.g., in between light and heavy keys). Assume relative volatilities are constant: αEB= 13.14 and αPB = 3.91.
a . Find Nmin
b . Find the fractional recovery of propane in the distillate.
D11. A saturated liquid feed that is 30.0 mole % ethanol and 70.0 mole % n-propanol is to be separated in a distillation column at 1.0 atm. The relative volatility of ethanol with respect to n-propanol is constant at 2.10 (ethanol is more volatile). We want the distillate to be 97.2 mole % ethanol and the bottoms to be 98.9 mole % n-propanol. Feed rate is 150.0 kmoles/hr. Use the Underwood equation to estimate (L/D)min.
D12.*a. 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 mole % benzene, 30 mole % toluene and 30 mole % cumene. The feed is input as a saturated vapor. We desire 99% recovery of the toluene in the bottoms and 98% 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 as the reference component, αB = 2.25 and αC = 0.210. 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. For the distillation problem given in part a, find the minimum reflux ratio by use of the Underwood equations. Use a basis of 100 moles of feed/hr. Clearly state your assumptions.
d . For L/D = 1.25(L/D)min, find the total number of equilibrium stages required for the distillation problem presented in parts a and b. Use the Gilliland correlation. Estimate the optimum feed plate location.
D13 * We have a column separating benzene, toluene, and cumene. The column has a total condenser and a total reboiler and has 9 equilibrium stages. The feed is 25 mole % benzene, 30 mole % toluene, and 45 mole % cumene. Feed rate is 100 moles/hr and feed is a saturated liquid. The equilibrium data can be represented as constant relative volatilities: αBT = 2.5, αTT = 1.0, and αCT = 0.21. We desire 99% recovery of toluene in the distillate and 98% recovery of cumene in the bottoms. Determine the external reflux ratio required to achieve this separation. If αBT = 2.25 instead of 2.5, how much will L/D change?
D14. Estimate the number of equilibrium contacts needed to separate a mixture of ethanol and isopropanol. The relative volatility of ethanol with respect to isopropanol is approximately constant at a value 2.10. The feed is 60.0 mole % ethanol and is a saturated liquid. We desire a distillate that is 93.0 mole % ethanol and a bottoms which is 3.2 mole % ethanol. Use an external reflux ratio, L/D that is 1.05 times the minimum external reflux ratio.
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 is to have a total condenser and a partial reboiler, and the optimum feed stage is to be used. Reflux is returned as a saturated liquid, and L0/D = 1.2. Feed rate is 1000 kg moles/hr. Feed is 39.7 mole % benzene, 16.7 mole % toluene, and 43.6 mole % cumene and is a saturated liquid. We desire to 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 Fenske-Underwood-Gilliland 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 kg moles/hr. Feed is 60 mole % ethanol and is a saturated vapor. We desire xD = 0.99 mole fraction ethanol and xB = 0.008 mole fraction ethanol. Reflux is a saturated liquid. There are 30 stages in the column. Use the Fenske-Underwood-Gilliland approach to determine
a . Number of stages at total reflux
b . (L/D)min
c . (L/D)actual
D17. A distillation column is separating toluene and xylene, α = 3.03. Feed is a saturated liquid and reflux is returned as a saturated liquid. p = 1.0 atm. F = 100.0 kg moles/hr. Distillate mole fraction is xD = 0.996 and bottoms xB = 0.008. Use the Underwood equation to find (L/D)min and Vmin 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 |Qc,min| and QR,min as toluene feed concentration increases?
D18. A depropanizer has the following feed and constant relative volatilities:
Methane (M): zM= 0.229, αM-P = 9.92
Propane (P): zP= 0.368, αP-P = 1.0
n-butane (B): zB= 0.322, αB-P = 0.49
n-hexane (H): zH= 0.081, αH-P = 0.10
Reflux is a saturated liquid. The feed is a saturated liquid fed in at 1.0 kg moles/(unit time). Assume CMO.
a .* L/D = 1.5, FRP,dist = 0.9854, FRB,bot = 0.8791. Estimate N.
b . N = 20, FRP,dist = 0.9854, FRB,bot = 0.8791. Estimate L/D.
c . Find the split of normal hexane at total reflux using Nmin.
d . L/D = 1.5, FRP,dist = 0.999, FRB,bot = 0.8791. Estimate N.
Note: Once you have done part a, you don’t have to resolve the entire problem for the other parts.
D19. Revisit problem 7.D4. Using the process simulator we found Nmin = 11 for this problem. Using this value plus the simulation data in 7.D4. part d, estimate (L/D)min using the Gilliland correlation.
D20. A distillation column is separating a mixture of benzene, toluene, xylene and cumene. The feed to the column is 5.0 mole % benzene, 15.0 mole % toluene, 35.0 mole % xylene and 45.0 mole % cumene. Feed rate is 100.0 kg moles/hr 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 = 0.330, and cumene = 0.210.
a . Find distillate and bottoms flow rates.
b . Find the number of equilibrium contacts at total 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 mole % nitrogen and 21.0 mole % 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.