Ind. Eng. Chem. Res. 1994,33,1397-1401
1397
Maximum Fractionation by Distillation of Systems with Constant Relative Volatilities Massimiliano Barolo and G. Berto Guarise**+ Zstituto di Zmpianti Chimici, Uniuersitb di Padova, Via Marzolo 9, 35131 Padova, Italy
This paper addresses the problem of shortcut simulation of a continuous distillation column. A computationally inexpensive approach is presented which considers the separation of a given feed in a column with an infinite number of stages in both the enriching and the stripping sections. The results so obtained represent the maximum fractionation that can be achieved for the given mixture at the given operating conditions. Reference is made to multicomponent systems with constant relative volatilities, and the classical Underwood equations are used. Examples are presented that show a good agreement with the results obtained in the case of a column with a limited number of theoretical stages. Due to its simplicity, the method appears to be an attractive tool for preliminary studies on distillation column design, operability, and control. Introduction
the infinite-stage column has been already adopted, for example, to describe the separation and the vapor requirements in batch distillation of binary mixtures (Bortolini and Guarise, 1970;Bauerle and Sandall, 1986). Reference is made to multicomponent mixtures with constant relative volatilities, and the method developed by Underwood (1946,1948) for the calculation of minimum reflux in the design problem is considered. The method does not hold for highly nonideal mixtures and should be regarded as a preliminary tool for rigorous distillation column design, operability,and control studies.
In the literature, a number of approximate and rigorous methods are available for the evaluation of product compositions in the continuous distillation of a given mixture (King, 1980;Henley and Seader, 1981;Holland, 1981;Kister, 1992). Most of these methods rely on stageby-stage calculations or simultaneous solution of the MESH equations, which can be combersome and time consumingeven with the current availability of computing capabilities. When accurate phase equilibrium and enthalpy data are lacking it is often not convenient to resort to rigorous thermodynamic models, since in many cases one would like to obtain just an approximate estimate of distillate and bottom composition, thus sacrificing rigorousness of computation to the speed of calculation. This is the case, for example, of the determination of initial guesses for iterative solution of rigorous optimization studies, and of the quick, first examination of the influence of the operating conditions on the fractionation of a given feed in a given column. Moreover, a rough estimate of the product composition changes in the face of disturbances in the loads and in the manipulated variables can be very useful for control purposes. Shortcut methods are known to represent a compromise between accuracy and simplicity that can be conveniently used in most practical cases. In fact, the design of a continuous distillation column by means of shortcut models is a well-established technique in chemical engineering (McCabe and Thiele, 1925;Fenske, 1932;Jenny, 1932;Gilliland, 1940;Underwood, 1946,1948;Hengstebeck, 1961;Jafarey et al., 1979). Besides the design problem, the usefulness of these simple models has been successfully proven also for continuousdistillation on-linecontrol (Cott et al., 1989), batch distillation design (Diwekar and Madhavan, 1991), and batch distillation simulation (Sundaram and Evans, 1993). This paper addresses the problem of the shortcut simulation of a continuous distillation column. A computationally inexpensive approach is presented which considers the separation of a given feed in a column with an infinite number of stages in both the enriching and the stripping sections. The results so obtained represent the maximum fractionation that can be achieved for the given mixture at the given operating conditions. The model of t
Problem Formulation Let us determine the maximum fractionation of a given feed rate F for given values of the base vapor and reflux flow rates (V’ and L, respectively). The problem is approached by considering the separation of the given feed in a column having an infinite number of plates in both the enriching and the stripping sections and operating at the given conditions (V’,L). This situation corresponds to the limiting fractionating capacity in the simulation problem as well as to the minimum reflux requirement for a given separation in the design problem. It is well-known that in each section of such a column at least one point is reached where only infinitesimal changes in component concentrations occur for successive plates (Holland, 1981). This point is known as a pinch point. The occurrence of pinch points allows an easy evaluation of the product compositions. With reference to the simulation problem we assume that the base vapor flow rate is fixed a priori;consequently the maximum separation of F is a function of the operating parameter L only. Therefore the problem may be formulated in this way: examine the influence of the reflux flow rate over the product compsitions X D , and ~ X B on ~ an infinite-plate distillation column. For this purpose, the thermodynamic isothermal work of separation of the feed in the two products can be assumed to give a measure of the fractionation ( C o d and Stuart, 1964). In dimensionless form:
Let us suppose that the following hypotheses hold: the relative volatility ai of component i is constant, the reflux is introduced at ita boiling point, the molal flows through-
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1398 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
out the enriching and stripping sections are constant, the overhead vapor Vis totally condensed. As for the system, let us assume that the relative volatilities, the feed composition Z F , ~ ,and the feed quality q are known. Limiting Conditions for the Reflux Flow Rate. A. Lower Limit. It corresponds to the condition D = D,, or B = Bmh. When the base vapor flow rate is fixed, two cases may occur in respect to the choice of the feed flow rate. In fact, the reflux flow rate can be set to zero (i.e., Lmin = 0)only in the case F L V ,where V = V’ + (1 - q)F. When F < V , a minimum flow rate Lmin= V - F must be returned into the column as reflux. B. Upper Limit. It corresponds to the condition D = Dmin = 0 whence B = B,, = F and Lm, = V. Model Development
According to Shiras et al. (1950), multicomponent separations in infinite-plate columns can be classified under two classes. Class 1 Separations. The pinch points for both sections are located a t the feed plate. Therefore, x:i
= xli = x F , i
the compositions of the pinch points are those corresponding to the feed flash (eq 2). Class 2 Separations. Equations 3 and 4 still hold, but (or x;) and zF,i is lacking. a direct correlation between The search of product compositions for a given reflux in an infinite-plate column is performed by using the method proposed by Underwood (1946,1948) for the calculation of the minimum reflux. It should be pointed out that in a simulation problem (or rating problem), such as the one considered here, the concept of key component looses meaning since the definition of the product compositions is the final result of the simulation. In this case, it is necessary to first determine which components are distributing and then determine the distribution of all the components. As a first guess of the number of components in the distillate and in the bottom (Nd and Nb, respectively) one can assume all Nc components are present in both products. Then
(2)
where x:i and xli are the mole fractions of component i in the enriching and stripping section pinch points, respectively, and X F , ~is the mole fraction of the same component corresponding to the feed flash. Material balances around the pinch points give
is solved for those 6 roots that lie between the ai values of the Ndb = Nd + N b - Nc distributing components. These roots (which are Ndb - 1 in number) are also solutions of the following equations: aixD,i
- -v
Z Z - D
(3)
(11)
and
and
Nc a i x B , i
(4)
where Kri and Kr’ are the equilibrium ratios of component i in the enriching and stripping section pinch points, respectively. To ensure that all the components of the feed distribute into the two products it is necessary that
VKFpc- L L 0
v’
---
Zai-8-
B
(12)
The composition constraints for the distributing components are Nd
Nc-Nb
Nd
Nc
(5)
and
and
L‘ - V’KF,,L 0
(6)
Components are ordered by decreasing relative volatilities from the most volatile one (component 1) to the least volatile one (component Nc). By combining inequalities 5 and 6, the following condition on the feed flow rate is obtained:
F?F=V’
- KF,Nc + (l- q)KFJyc KF,l
(7)
Since inequalities 5 and 6 must hold independently, we also get the condition L1* I L IL2*, where
L1* = V‘KF,,- qF
(8)
and
Thus, providing that the conditions F 2 F and LI* I L I L2* hold, the maximum separation can be determined by solvingeqs 3 and 4 with respect to X D , and ~ x B , i , whereas
where X D ~= F z F ~ J D and X B = ~ FZFj/B refer to the nondistributing components. The concentrations of the distributing components in the products are determined by solving the two linear systems of eqs 11 and 13 for the distillate and eqs 12 and 14 for the bottom. In both cases the number of equations and unknowns is Ndb. By checking that the X D , ~and X B , values ~ from the above systems fall in the range 0-1, Nb and Nd are adjusted and the procedure is iterated until convergence is reached. The number of iterations is normally less than 3. It should be noted that eqs 11 and 12 cannot be used when Ndb = 1is guessed. In this case the mole fraction of the distributing component in the products is evaluated simply by means of the mass balance of this component and the corresponding constraint conditions (13 or 14). Since the assumption of Ndb = 1 still needs to be checked, one should guess the presence in the distillate of the next (lower ai value) component “on the right”, so that eq 11
Ind. Eng. Chem. Res.,Vol. 33, No. 5, 1994 1399
can be applied for the 0 value lying between the ai of these two components according to the above procedure. If the mole fraction of the adjoined component is sound, then that component is actually present in the distillate and Ndb = 2. The same check is performed for the first component “on the left” (higher ai value) in the bottom. As for the determination of the compositions of the enriching section pinch point, the following expression is derived from eq 11and from the material balances around the enriching section:
Nd
L=V-
CFZ,,~
(18)
111
The base vapor flow rate is obtained from eq 17 for the same e* value or simply as V’ = V - (1- q)F. It can be shown (King, 1980) that r
L
Nd
e* = -
v r=l
as well as
Equation 15 is satisfied when one of the factors becomes zero. It is easy to show that the term in brackets is less than zero when Nd > 1. So substituting into eq 15the Nd - 1solutions of eq 11and solving the system of the pinch point composition constraint and of the Nd - 1equations which are obtained by setting to zero the second term of eq 15, one gets the desired xii. A similar procedure is developed for the stripping section. It is worth noting that the computational demand of the whole procedure is very little and is almost entirely related to the solution of the Underwood equations. Once the roots of eq 10 have been determined, the computer time can be further reduced by calculating the feed flash without using standard iterative subroutines. In fact, from eq 10 and from the material balances of the feed flash one gets an expression formally analogous to eq 15, but where the slope of the q-line, i.e., - q / ( l - q), replaces the slope of the operating line, Le., LIV, and xF,i replaces x ; ~The liquid phase composition is then evaluated by solving the system of its constraint equation and of the Nc - 1 equations derived by setting to zero the second factor of the above eq 15. Case Ndb = 0 (No Distributing Component). The case of no distributing component represents a possible situation for operating columns. The knowledge of the operating conditions for the occurrence of such splits would be helpful, for example, in the preliminary solution of sequencing problems. A similar problem has been addressed by Glinos and Malone (1984), who presented simple algebraic expressions which allow straightforward calculation of minimum reflux for splits of a saturated multicomponent feed. The operating conditions for Ndb = 0 can be obtained by considering eqs 11 and 12 in the form
(17) where e* is the root of eq 10 lying in the interval CYNd+lThe existence of a common root in eqs 10, 16, and 17 is due to the fact that eq 10 can be obtained as a sum of eqs 16 and 17 with the same e* value. Therefore the VIF ratio for “cutting” the feed between the components Nd and Nd + 1 is obtained from eq 16, and the corresponding reflux flow rate is then evaluated as
Equations 19and 20 allow one toevaluate the compositions of the pinch points according to eqs 3 and 4 in the form
i = 1, ...,Nd Fz,,~= L’x;( 1-
s)
i = Nd
+ 1,...,Nc
(22)
respectively. Examples
(A)The effect of varying the reflux flow rate on the separation of a ternary mixture with a1 = 3, a 2 = 2, and a 3 = 1 is considered. Feed characteristics are ZFJ = 0.3, Z F , 2 = 0.3, Z F , ~= 0.4,and q = 1. At the feed composition the Ki values are K I = 1.579, K 2 = 1.053, and K3 = 0.526. The limiting base vapor flow rate per unit feed that gives the possibility that all the components are present in both products (class 1 separation) is found by eq 7. It results in V’IF = 0.95. Figure 1compares the product compositions obtained by solving the maximum fractionation problem (lines) to those (points) obtained in a 20-stage column with the feed in the central position ( N = N’ = 10); both simulations were run with V’IF = 1. It is shown that the approximation reached with the proposed procedure is indeed satisfactory. The approximation becomes even better when V‘IF is reduced, as can be seen from Figure 2, where dt is plotted versus the normalized reflux flow rate ( L - L-)/(L- Lmin) for two different values of the V’IF ratio. Of course, dt > 0 as long as the flow rate of both the fractionation products are greater than zero. (B)The gasolinestabilization is one example considered by Gilliland (1940). Feed composition and relative volatilities are listed in Table 1. The following conditions are specified: propane mole fraction in the residue as low as 0.0025 and 96 9% butane recovery in the same product (Robinson and Gilliland, 1950);Gilliland reports a minimum reflux ratio of about 0.53. By the Underwood method the product compositions given in columns 4 and 5 of Table 1are calculated, corresponding to a minimum reflux ratio of 0.5726. The same values are found when considering the maximum fractionation at the operating conditions LIF = 0.1795 and V’IF = 0.9131 (q = 1.42). These values are compared with the results given by a rigorous simulation (ai = constant) of the separation obtained in different columns (with the feed in the central position). The agreement with the separation of a 60-stagecolumn is complete, but it can be considered good even in the case of a 20-stage column.
1400 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 Table 1. Gasoline Stabilization Example: V'/F = 0.9131; L / F = 0.1795. N=N'-m N = N' = 10 compon CH4 C2H6
C3H6
C3He i-CdHlo
n-Cd C6
C6
c7 C8
residue
a
ZF
XD
XB
XD
XB
XD
XB
XD
XB
0.020 0.100 0.060 0.125 0.035 0.150 0.152 0.113 0.090 0.085 0.070
0.0638 0.3189 0.1913 0.3932 0.0136 0.0192 0 0 0 0 0
0 0 0 0.0025 0.0448 0.2098 0.2214 0.1646 0.1311 0.1238 0.1020
0.0638 0.3189 0.1891 0.3833 0.0179 0.0270 0 0 0 0 0
0 0 0.0010 0.0070 0.0428 0.2062 0.2214 0.1646 0.1311 0.1238 0.1020
0.0638 0.3189 0.1911 0.3931 0.0136 0.0194 0 0 0 0 0
0 0 0.0001 0.0025 0.0448 0.2096 0.2214 0.1646 0.1311 0.1238 0.1020
0.0638 0.3189 0.1913 0.3932 0.0136 0.0192 0 0 0 0 0
0 0 0 0.0025 0.0448 0.2098 0.2214 0.1646 0.1311 0.1238 0.1020
1.7626
8.0504
2.4094
1.3986
2.7972
1.2580
0.9691
0.9588
(11
= 4, a2 = 3, as = 2, a4 = 1, zp.1 = zp2 = z p =~ zp.4 = 0.25, and q = 0.5
15 50
0.8985
m
1 0.4996 0.5000 0.5000 0.3333 0.3333 0.3333
0.1013 0.0100 0 0.4665 0.4987 0.5000 0.3333 0.3333 0.3333
0.9900
15 50 m
15 50 m
a
N=N'=30
26 6 2.6 2.3 1.23 1 0.43 0.18 0.08 0.03 0.005
Table 2. Ndb = 0. Conditions of the Feed 3.5580
N = N ' = 20
0.0002 0 0 0.0339 0.0013 0 0.3248 0.3332 0.3333
0 0 0 0 0 0 0.0085 0.0002 0
0.0338 0.0033 0 0.0004 0 0 0 0
0.2996 0.3300 0.3333 0.0335 0.0013 0 0 0 0
0
0.3333 0.3333 0.3333 0.4661 0.4987
0.3333 0.3333 0.3333 0.5000 0.5000
0.5OOo
0.5OOo
0.0266 0.0005
0.9744 0.9995 1
0
Equation 10. b Equation 17. 0.3
0.2
0.6 X
X
0.4
0.1
0.2
0.0
0.0
0.0
0.2
0.8
0.6
0.4
0.0
1.0
L/F
-
I
V'/F-l.O
-7
A
I
0.5
0.3
0.4
0.5
0.2 X
0.1
4
I
0.3 -
0.4
I
I
0.4
0.3
0.3
I
P.
0.2 L/ F
Figure 1. Product composition as a function of reflux flow rate in the case V' > V'* and comparison between column with N , N' (lines) and N = N' = 10 (points). System: a1 = 3, a2 = 2, as = 1, ZFJ = Zp,) = 0.3, q = 1, and base vapor relative flow rate V'IF = 1. 0.5
0.1
i-
4, 0.2
J
0.0 0.0
T
0.0
I
0.2
0.4 0.6 (L-Lmdl(LmaiLmin)
I
0.8
0.2 L/F
\ 0.0
0.1
'
1.0
Figure 2. Maximum separation as a function of the ratio (L - L,&/ ( L , - L-) for two different V'IF ratios. System and columns as in Figure 1.
Figure 3 shows the trajectory of the bottom composition in the whole reflux range as obtained both with the infinite stage model (Figure 3a) and with the simulation of a 20stage column (Figure 3b). Again the results are very close in the entire reflux range as it is confirmed in terms of dt (Figure 4).
-
Figure 3. Gasoline stabilization example: bottom composition as a function of the ratio LIF in the cases (a) N , N' and (b)N = " = 10.
(C) The operating conditions for getting Ndb = 0 for the following feed are calculated: Ne = 4, a1 = 4, CYP = 3, CY3 = 2, CY4 = 1, ZFJ = Z F , ~ Z F , ~= Z F , ~= 0.25, and Q = 0.5. The results are collected in Table 2. In the table the product compositions are also shown for a different number of equilibrium stages and feed in the central position as resulting from a rigorous simulation. It is evident how the increase in the number of stages brings the product compositions toward those of the infinite-plate column characterized by the sharp separation.
Ind. Eng. Chem. Res., Vol. 33,No. 5,1994 1401
V, V‘ = vapor mole flow rate in the enriching and stripping section, respectively
V’* = characteristic vapor mole flow rate defined by eq 7 for a given F x , y, z = mole fraction in saturated liquid, saturated vapor,
and generic, respectively a = relative volatility 0.2
4
& = maximum fractionation in terms of thermodynamic isothermal separation work (eq 1) 0 = parameter of the Underwood eqs 10,11, and 12
0.0
Subscripts B, D,F = bottom, distillate, feed, respectively i, j = component e, s = enriching and stripping section, respectively
0.0 0.1
0.3
0.2
0.4
0.5
L/ F
Figure 4. Gasoline stabilization example: comparison of the maximum separation (line)with the separation obtained in a column with N = N’= 10 (pointa).
The effect of the feed quality can be evidencedby solving the same problem with a saturated liquid feed (q = 1). One gets LID = 6.855 (Nd = l), 2.421 (Nd = 2), and 0.837 (Nd = 3) Note that in this latter case the method of Glinos and Malone (1984)also applies. Conclusions
A method has been proposed that allows the determination of the maximum fractionation of a given multicomponent ideal mixture at given operating conditions. The method relies on the Underwood model for minimum reflux calculation. The separation obtained with this shortcut model compares well with that obtained in a column with a limited number of theoretical stages. Although the method does not hold for highly nonideal mixtures, ita simplicityand computational inexpensiveness qualify it as an attractive tool for preliminary studies on distillation column design, operability, and control. Acknowledgment
The financial support granted to this work by the Italian National Research Council (CNR, Progetto Finalizzato Chimica Fine) is gratefully acknowledged. Nomenclature B = bottom mole flow rate D = distillate mole flow rate F = feed mole flow rate F = characteristic feed flow rate defined by eq 7 for a given
V’ K = vapor-liquid equilibrium constant L,L’ = liquid mole flow rate in the enriching and stripping section, respectively
L1*, L2*= characteristic reflux flow rate defined by eqs 8 and 9,respectively
N,N‘ = number of theoretical
stages in the enriching and stripping section, respectively N b = number of components in the bottom Nc = number of componenta of the system Nd = number of componenta in the distillate N d b = number of distributing components q = feed quality
Superscript 01 = pinch point
Literature Cited Bauerle, G. L.; Sandall, 0. C. Batch Distillation of Binary Mixtures at Minimum Reflux. MChE J. 1987,33,1034-1036. Bortolini, P.; Guarise, G. B. A New Practice of Batch Distillation (in Italian). Quad. Zng. Chim. Ztal. 1970,6,150-158. Cott, B. J.; Durham, R.G.; Lee, P. L.; Sullivan, G. R. Process ModelBased Engineering. Cornput. Chem. Eng. 1989,13,973-984. Coull, J.; Stuart, E. B. Equilibrium Thermodynamics; Wiley: New York, 1964;p 380. Diwekar, U. M.; Madhavan,K. P. MulticomponentBatch Distillation Column Design. Znd. Eng. Chem. Res. 1991,30,713-721. Fenske, M. R.Fractionation Straight-Runof Pennsylvania Gasoline. Znd. Eng. Chem. 1932,24,482-485. Gilliland, E. R. Multicomponent Rectification-Minimum Reflux Ratio. Znd. Eng. Chem. 1940,32,1101-1106. Glinos, K.; Malone, M. F. Minimum Reflux, Product Distribution, and Lumping Rules for Multicomponent Distillation. Znd. Eng. Chem. Process Des. Dev. 1984,23,764-768. Hengstebeck, R.J. Distillation-Principles and Design Procedures; Reinhold Publishing: New York, 1961. Henley,E. J.; Seader,J. D. Equilibrium-Stage Separation Operations in Chemical Engineering; Wiley: New York, 1981. Holland, C. D. Fundamentals of Multicomponent Distillation; McGraw-Hill: New York, 1981. Jafarey, Y .;Douglas, J. M.; McAvoy, T. J. Short-CutTechniquesfor Distillation Column Design and Control. 1. Column Design. Znd. Eng. Chem. Process Des. Deu. 1979,18,197-202. Jenny, P. J. Graphical Solution of Problems in Multicomponent Fractionation. Trans. Am. Znst. Chem. Eng. 1939,35,636-677. King, C. J. Separation Processes, 2d ed.; McGraw-Hill: New York, 1980. Kister, H.2.Distillation Design; McGraw-Hill: New York, 1992. McCabe, W. L.; Thiele, E. W. Graphical Design of Fractionating Columns. Znd. Eng. Chem. 1925,17,605-611. Robinson, C . S.R.;Gilliland, E. R. Elements of Fractional Distillation, 4th ed.; McGraw-Hill: New York, 19M); p 261. Shiras, R.N.; Hanson, D. N.; Gibbson, C. H. Calculation of Minimum Reflux in Distillation Columns. Znd. Eng. Chem. 1960,45,871876. Sundaram,S.;Evans, L. B. ShortcutProcedure for Simulating Batch Distillation Operations. Znd. Eng. Chem. Res. 1993,32,511-518. Underwood, A. J. V. Fractional Distillation of Multi-Component Mixtures - Calculation of Minimum Reflux Ratio. J. Znst. Petrol. 1946,32,614-626. Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures. Chem. Eng. Prog. 1948,44,603-613. Received for reuiew September 13, 1993 Revised manuscript received February 3, 1994 Accepted March 2, 1994. Abstract published in Advance ACS Abstracts, April 1,1994.
