Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 1087-1090
L. W. Heavner, P. M. Shain, and R. Terri. Registry NO. CaO, 1305-78-8;C6HhCH3, 108-88-3;C&, 7143-2; C, 7440-44-0; 1-methylnaphthalene,90-12-0; n-heptane, 142-82-5.
Literature Cited Appleby. W. G.; Avery, W. H.; Meerbott, W. K. J. Am. Chem. SOC. 1947, 69, 2279. Bradbury, A. G. W.; Shafizadeh, F. Csrbon 1980, 78, 100. Ellig, D. L. M.S. Thesis, Department of Chemical Engineerlng, Massachusetts Institute of Technology, Cambrldge, MA, 1981. Elovich, S. Y.; Zhabrova, G. M. Z h . Fiz. Khim. 1939, 73, 1761. Fitzer, E.; Mueller, K.; Schaefer, W. in "Chemistry and Physics of Carbon"; Walker, P. L., Jr., Ed.; Marcel Dekker: New York, 1971; Vol. 7, pp 237-383. Johns, I.B.; McElhill, E. A.; Smith, J. 0. J. Chem. Eng. Data 1982, 7 , 277. Longwell, J. P.; Krasniak, S.; Lai, S. C. K.; Williams, G. C.; Peters, W. A. Technical Progress Report FE-MIT-30229-2 to the US Department of Energy, Morgantown Energy Technology Center, 1981. Longwell, J. P.; Lai, S. C. K.; Williams, G. C.; Peters, W. A. Technical Progress Report FE-MIT-30229-3 to the US Department of Energy, Morgan-
1087
town Energy Technology Center, 1982. Madison, J. J.; Roberts, R. M. Ind. Eng. Chem. 1958, 5 0 , 237. Mead, D. W. M.S. Thesis, Department of Chemical Engineering, Massachusetts Instmute of Technology, Cambridge, MA, 1979. Nickels, J. E; Corson, B. 8. Ind. Eng. Chem. 1951, 43, 1685. Rice, F. 0.; Johnson, W. R. J. Am. Chem. SOC. 1934, 56, 214. Satterfield, C. N. "Heterogeneous Catalysis in Practice"; McGraw Hill: New York, 1980; p 318. Schachter, Y.; Pines, H. J. Catal. 1988, 1 7 , 147. Snow, M. J. Department of Chemical Engineering, Massachusetts Instltute of Technology, Cambridge, MA, Personal Communlcatlon. 1981. Szwarc, M. J. Chem. Phys. 1948, 76, 128. Yeboah, Y. D. Sc.D. Thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1979. Yeboah, Y. D.; Longwell, J. P.; Howard, J. 8.; Peters, W. A. Ind. Eng. Chem. Process D e s . Dev. 1980, 79, 848. Yeboah, Y. D.; Longwell, J. P.; Howard, J. B.; Peters, W. A. Ind. Eng. Chem. Process Des. Dev. 1982, 27, 324.
Received for review April 8, 1983 Revised manuscript received February 4, 1985 Accepted February 13, 1985
Minimum Vapor Flows in a Distillation Column with a Sidestream Stripper Konstantlnos Gllnos and Mlchael F. Malone" Department of Chemical Engineering, Goessmann Laboratory, University of Massachusetts. Amherst, Massachusetts 0 7003
A shortcut procedure to evaluate the minimum vapor flows in distillation columns with a sidestream stripping section is presented. The method Is simple and highly accurate for relatively ideal mixtures. The case of a ternary mixture is discussed, but the algorithm readily extends to mixtures with any number of components. Also, the approach can be generalized to other types of complex columns as well.
Complex distillation columns have been shown to result in substantial savings in both operating and capital costa in various cases examined in the literature (e.g., Petlyuk et al., 1965; Cerda and Westerberg, 1979). Two typical types of complex columns have sidestream strippers or rectification sections, e.g., side-stripper columns have been traditionally used in the petroleum industry. Estimation of the vapor rates in distillation is essential in order to size, cost, and compare the various design alternatives. However, there are no simple methods currently availale in order to estimate the minimum vapor flows required in complex columns. Cerda and Westerberg (1979) presented a shortout method, but we think that the algorithm is rather complicated. We present here a much simpler method to evaluate the minimum vapor flows in sidestripper columns. The work that follows is based on an ideal ternary mixture but can easily be extended to ideal multicomponent mixtures. An analogous treatment of the sidestream-rectifier column is also straightforward. Met hod A side-stripper distillation column separating the ternary mixture A, B, and C is shown schematically in Figure 1. Two separations take place: A from B in sections 1 and 2, and B from C in sections 3 and 4. Both of the splits can be as sharp as is desired. If all the external flows are fixed, there remain two internal flows that have to be regulated in order to operate the column. That is, instead of having to fix a single internal flow as a design variable, as we do in a simple distillation column performing a single sepa0196-4305/85/1124-1087$0 l.50/0
ration task, we have to fix two internal flows as design variables. Consequently, two prospective pairs of pinch zones appear in this column. One of them is around the side-stripper tray (between sections 1 and 3 in Figure l), and the other is around the feed tray. We will evaluate the minimum vapor rates required in the side-stripper column in order to perform the separations to the desired extent, when infinite trays are assumed in all four pinch zones. The method presented is based on Underwood's equations (Underwood, 1946, 1948). Therefore, constant relative volatilities and molal overflows in each column section are assumed. In Figure 2, an equivalent representation of the sidestripper column is drawn. The complex column is broken into two simple columns. The feed enters the second column (sections 3 and 4), and the first column serves as a kind of partial condenser for the second. The feed to the first column is saturated vapor (stream V3)and there is a liquid sidestream exiting the first column (stream L3) about one tray above or below the side-stripper tray (which is the feed tray of column 1). The reflux ratio for the first column is R1 = L J D , while the reflux ratio for the second column is defined as R2 = L,/(P + D). The minimum vapor rate in column 2 can be found from Underwood's equations, if the distillate component flows are considered as the net upward flows of these components at the point where V is evaluated (King, 1980). If we evaluate (V,),, that is, the minimum vapor rate rising up from the feed plate, then 0 1985 American Chemical Society
1088
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
i
1 3 1
aw
Figure 2. Equivalent representation of a column with a sidestream stripping section.
Xi,w
Figure 1. Distillation column with a sidestream stripping section.
(V3)m
=
~A(DXA,D + ~ X A , P )QIB(DXB,D + ~XB,P) ffA
+
-
+
ffB -
+ pxC,P)
(DxC,D
1-E
(1)
where f is one of the roots of the equation
~ A X A , F~ B X B , FXC,F ffA-E
+- f f B - f
l-E
In column 2, the separation B/C takes place. Therefore, the root of eq 2 we have to use in (1) satisfies aB> Ez > 1 (Underwood, 1946). Throughout the rest of this text, the various roots of eq 1are ordered so that t3> E2 > f l . Consider the first column and assume that the liquid sidestream (stream L3 in Figure 2) leaves the column one or two trays above the side-stripper tray. Underwood’s equations for minimum reflux are (Vl)m =
ffA(DXA,D ffA
+ L3XA,3) - c;‘
+ ffB(DXB,D+ L3XB,3) + ffB
- E‘
(DxC,D + L3xC,3) 1-E‘
(3)
In this column, the A/B separation is performed. Therefore, c;‘ is that root of the following equation, which satisfies CYA > c;‘ > aB,and we denote it by E3/ “AYA.3
ffBYB.3
yC.3
where Yi,3is the composition of stream V, which is the feed to the first column. Also, since V, is assumed to be saturated vapor, q = 0. Moreover, the compositions of streams L3 and V, are coupled by the mass balance Yi,, = @Xi,, pxi,p + L3Xi,,)/v3 i = A, B, c ( 5 ) Although eq 3,4, and 5 are coupled and tedious to solve, they can be greatly simplified as follows. Under limiting flow conditions throughout the side-stripper column, the upper pinch zone of column 1will be located at, or a few trays above, the side-stripper tray, depending on the amount of the heavy nonkey component C entering with the feed V3. For relatively sharp separations, this amount
+
ffAXA,3
ffA-F
(2)
+--1-q
will be small, and one or two trays will be enough to wash C out of the distillate product. If the separation B/C is really sharp, this pinch will occur at the side-stripper tray. Therefore, the composition of stream L3 coincides approximately with the pinch composition. We know that the latter satisfies the equation (Underwood, 1946)
+-ffBxB,3 ffB-F
xC,3
+-=
1-Q
0
(6)
There are three different sets of pinch compositions, each one corresponding to a different root c;‘ of eq 3. For a given separation task, only one of them has physical significance. Underwood (1946) proves that if component B is the heavy key, the composition reached in the upper pinch zone is that corresponding to the root &‘ which satisfies aB> &‘ > 1. He also proved that if eq 6 is written for this pinch composition, it is satisfied by both roots [2/ and f3/ of eq 3, where cyA > f3/ > LIB. Therefore, eq 3 in view of eq 6 simplifies to ~ A D X A , D~ @ X B , D DXC,D - (Vl)m (7)
+
ffA-E3
ffB-E3
+--
- l 3
Also, we can replace Yi,3from eq 5 in eq 4. We find (rhs of eq 1) + L3(lhs of eq 6) = (V3)m (8) The second term above becomes zero for c;‘ = 54, and eq 8 reduces exactly to eq 1. Therefore, E,’ is a root of eq 1 as well (it is known to be a root of eq 3,4, and 6 already) and f 3 = f i . This result is very important, because the procedure to evaluate the minimum vapor rates in a side-stripper column now becomes straightforward. However, we have to solve two higher order equations, that is eq 1for E2 and eq 2 for f 3 . The algorithm is as follows: (1) Solve eq 2 for l2where > t2> 1. (2) Find (VJ,,, from eq 1for the above value of 5. (3) Solve eq 1for its root from eq 7. f 3 , CYA> f 3 > CYB. (4) Find We have assumed that the liquid draw off from the first column, L3,is taken off one or two trays above the sidestripper tray. We can repeat the whole analysis assuming that L, is taken off below the side-stripper plate; that is, we would assume that the composition of stream L3 is the lower pinch zone composition. What we find in this case is that we arrive at exactly the same results and equations we presented above, so that the design of the column is independent of whether we take the liquid draw off above
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 1089 EFFECTIVE DISTILLATE FLOW = V, - L,
or below the side-stripper plate. Physical Interpretation of the Above Results If we divide both sides of eq 1by ( P + D ) , we can write
NET FEED: V - L
V
where with Xi,s we denote the fictitious composition Xj,s = ( DXj,D + PXi,p)/(P + D ) = (V3Yi,3- LJi,J/(V3 - ~ 5 3 ) (10) The root 5 of eq 9 which is between the relative volatilities of the keys CIA and must be used in eq 7 in order to find ( VJm. Therefore, eq 1 or its equivalent, eq 9, are actually the Underwood’s equations for the matching at the feed plate; that is, eq 9 is the analogue of eq 2 for column 1. Therefore, we can evaluate the limiting flows in column 1 of Figure 2 by assuming a feed flow equal to the net amount entering the column ( V3- L3)along with the feed compositions given by eq 10 and feed quality q = -(R2)m = -L3/(V3 - L3). This approach has been used before as an approximation, but here we prove that it is exact, for the case examined. This result is important because it generalizes Underwood’s equations to sections of complex columns with liquid return in approximate equilibrium with the vapor feed as the one shown in Figure 3a. Many complex columns fall into this category, e.g., columns with side sections and two types of Petlyuk columns, shown in Figure 3b (Petlyuk et al., 1965). Simplification of the Method for Sharp Separations If the B/C split is sharp, then the quantity DXc,D+ PXc,pin eq 1will be very small and the third term of this equation can be omitted. Now we have a quadratic with roots t2and t3. E2 is known because it is the root of (2) we calculated to find (V3)m. E3 is the root needed to evaluate ( VJm from eq 7, which can be easily found from the formula for the product of the roots of a quadratic equation, as
In this way, we avoid the solution of eq 1 for the root t3. Moreover, since XC,?is assumed to be negligible, eq 1with (2) can be written in the simple form
In addition, if the split A/B in column 2 is also sharp, the second and third terms of eq 7 are negligible. If in this expression we substitute t3from eq 11, we find
On the basis of the above, the algorithm to calculate the limiting flows is simplified to the following: (1)Find the root E of eq 2 which is aB> Ez > 1. (2) Find (V3)mfrom eq 12. (3) Find (VJm from eq 13. The solution of eq 2 remains iterative in general. However, any solution procedure converges in a few iterations, because the root is bounded. If q = 1or q = 0, eq 2 reduces to a quadratic, and eq 12 and 13 can be combined to give
Figure 3. (a) Portion of complex column with liquid return in approximate equilibrium with the vapor feed. (b) Two types of Petlyuk columns.
This simplified procedure holds only for ternary mixtures. But we feel that the full algorithm presented earlier has some generality for multicomponent mixtures because its physical interpretation makes sense regardless of the number of components. Example An example proposed by Stupin and Lockhart (1971) and solved by Cerda and Westerberg (1979) consists of a saturated liquid feed with compositions and volatilities given by components F A (LK) 0.333 B(MK) 0.334 C(HK) 0.333
D 0.333
P
B
‘3
0.333
9 3 1
0.334
With the simplified algorithm suggested (we assume very sharp splits), we find (per mole of feed) (VJm = 1.254 mol and (VJm = 0.991 mol. The method of Cerda and Westerberg gives (Vl)m= 1.252 mol and ( V3)m= 0.976 mol. We see that these numbers are in very close agreement. Generalizations The method we described and the algorithm proposed can be very easily generalized for ideal mixtures with any number of components, N. It is enough to write Underwood’s equations (I), (2), and (7) with N terms instead of 3. Note also that the method is not restricted to saturated liquid feeds. However, the simplified equations (11)-(13) hold true only for ternary mixtures.
1000
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
Acknowledgment This work was made with the support of the us D ~ partment of Energy (Grant DE-AC02-81ER10938). Nomenclature D = distillate flow rate, mol/h P = side-stripper product flow rate, mol/h W = bottom flow rate, mol/h F = feed flow rate, mol/h ( V J m= minimum vapor flow rate in column section i, mol/h L, = liquid flow rate in column section i, mol/h (I?& = minimum reflux ratio at top of column section 3 q = feed quality X I , = mole fraction of component i in liquid stream L, Y!,,= mole fraction of component i in vapor stream V,
Greek Symbols
relative volatility of component i with respect to the heaviest component present in the feed f i = roots of Underwood’s equations Literature Cited ai =
Cerda, J.; Westerberg, A. W. Paper presented at the 72nd AIChE Annual Meetlng, San Francisco, CA, 1979. King, C. J. “Separation Processes”, 2nd ed.;McQraw-Hill: New York, 1980. Petlyuk, F. B.; Platonov, V. M.; Slavlnskii, D. M. Int. Chem. Eng. 1965, 5 , (31, 55. Stupin, W. J.; Lockhart, F. J. Paper presented at the 64th AIChE Annual Meeting, Sen Francisco, CA, 1971. Underwood, A. J. V. J. Inst. Pet. 1846, 32, 614. Underwood, A. J. V. Chem. Eng. Prog. 1948. 4 4 , ( 8 ) , 603.
Received for review August 6, 1984 Revised manuscript received February 19, 1985
