I n d . E n g . C h e m . R e s . 1987, 26, 1839-1845
1839
Approximate Design of Multiple-Feed/Side-Stream Distillation Systems Ioannis P. Nikolaides and Michael F. Malone* Chemical Engineering Department, Goessmann Laboratory, University of Massachusetts, Amherst, Massachusetts 01003
An approximate, equation-based design method is presented to describe the separation of multicomponent mixtures in multiple-feed towers with or without side streams. Simple expressions for the selection of the controlling feed and the determination of feasible side-stream compositions are found. Several counterintuitive results are found. It is shown that the minimum reflux, rather than the boiling point, is a better criterion for ordering feeds t o the column. For example, vapor rates are sometimes reduced by placing lower boiling point feeds below higher boiling point feeds in multiple-feed columns. Complex configurations, using multiple-feed and side-stream columns, can reduce the operating costs for distillation without large increases in capital costs. Examples are shown with vapor-rate savings as large as 30%. These complex column arrangements may be useful in process retrofits. When streams of different composition are separated by distillation in a single tower, there is always some potential for a reduction in the vapor rate if the feeds enter the column a t different points. Multiple-feed columns appear in complex distillation designs such as those proposed by Petlyuk et al. (1965) and Elaahi and Luyben (1983). Side-stream columns can also be used in similar designs to reduce costs, when the desired side-stream purity can be attained without a large increase in vapor rate above that required for the primary separation. An approximate design method for side-stream towers with a single feed was described by Glinos and Malone (1985). Here, we describe a method for multiple-feed towers, with or without side streams, and show some of the potential economic advantages. The procedure is useful for mixtures with approximately constant volatility to the same extent as the development of Underwood (1946, 1948). The purpose of the methods described here is the evaluation of preliminary designs, the quick screening of alternatives, and the rapid estimation of the sensitivity of the design to the data. In the first section we consider the design of a multiple-feed tower without side streams for the common cases of binary and ternary mixtures, including general criteria to decide the “controlling” feed. In the second section, a procedure for the design of a multiple-feed column with side streams is described. Finally, the advantages of these systems are demonstrated by examining the separation of a ternary mixture using a complex column configuration and comparing it to traditional schemes.
Multiple-Feed Columns without Side Streams In a single-feed distillation column separating two “key” components in a mixture with approximately constant volatility, as the reflux ratio decreases to the minimum, two zones of constant composition appear, one above and one below the feed tray. These “pinch zones’’ are regions that would require an infinite number of trays to attain the specified separation. A multiple-feed distillation column with n feed streams will produce n pairs of such pinches. However, only one of these pairs is meaningful for the design of the column. As the reflux ratio decreases, the feed for which the first pair of pinch zones is established determines the actual minimum external reflux ratio for the column. For a binary mixture, this can be visualized easily on a McCabe-Thiele diagram, e.g., Yawns et al. (1981). However, it is not clear for a multicomponent mixture which
feed produces the first pinch zone, and one design procedure is to assume that each feed does and then to compare the corresponding reflux ratios. This can be done easily by examining each feed for an equivalent single-feed column. When the fth feed (numbered from the top of the column) is considered, the distillate and bottoms flows in the equivalent single-feed column are D - E{Z1lF,and B - Ejn_f+lF,,respectively. The controlling feed is that for which the minimum reflux is the largest, j = 1,2, ..., n R, = max [(R,),] (1) According to Underwood (1946, 1948), the minimum reflux ratio for a single-feed column can be estimated from
where 0 is the root of the polynomial ffixF,i
E ai _,=l-n
i-1
(3)
lying between the volatility of the heavy and the light keys. Similarly, for the external reboil ratio (4) The analysis can also be formulated for the multiple-feed tower, as follows. Consider the fth feed; for the ith component, the net upward flow of product i above this feed is Dx - x(21FjxF,,L, and the effective distillate rate is D - X$:F,. The minimum internal reflux ratio for this feed can be written by analogy with eq 2 as
With reference to Figure 1,rm,f can be defined as ‘mi
=
D-
EF~
(6)
j=1
The total liquid flow rate to the feed tray is related to the feed flow rate by f-1
L f = Lo
+ JC= 1q j F j
08S8-5S85/87/2626-~S39$01.50/0 0 1987 American Chemical Society
(7)
1840 Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987
where x is the composition of the light component in the j t h feed. If the feed stages are selected so that x f 1 x, for j = 1, 2 , ..., f-1 (e.g., physically this means that the feeds are fed from top to bottom in order of decreasing composition of the light component), the controlling reflux ratio is always less than the reflux ratio in the absence of all the other feeds. The magnitude of the difference depends on the feed conditions as shown in eq 13. For a double-feed column, a necessary and sufficient condition for the lower feed to control, i.e., R,,z > Rm,l,is simply
Fz - > - ax1 F1 1 - XZ
Figure 1. Multiple feed tower, showing the internal and external flows.
where Lo is the liquid external flow rate. From eq 6 and 7 with L o / D as the external reflux ratio, R,,/, we get
Equations 5 and 8 can be combined to give the final expression for the overhead reflux ratio
(9) where 0, is the root of eq 3 (written for the feed compositions of the fth feed from the top) lying between the relative volatilities of the key components. Equation 9 can also be written
where R;,/ denotes the reflux ratio of the fth feed in the absence of all the others; R,,f = RL,f for the first feed from the top of the column. Simple approximate expressions for the values of RLf are available for saturated liquid feeds from the work of Glinos and Malone (1984). An approximate solution for the reflux ratio for feeds other than saturated liquids is presented in the Appendix. A similar analysis for the stripping section leads to an expression for the minimum external reboil ratio, S,
(14)
where x1 and xz are the mole fractions of the light component in the two feeds. Even for this simple case, the actual reflux ratio depends not only on the feed flows and feed compositions but also on the relative volatility. Although it may seem intuitive that the controlling feed is the one with the smallest flow of light component, especially when the flow rates are similar, this is not the case unless the separation is extremely difficult ( a = 1). It is also of interest to estimate the potential vapor savings from using a double-feed column instead of premixing the streams. If the second feed from the top is controlling, the savings in the minimum vapor rate is Av, =
(-)(
x1 - x2
CY-1
5 , 1-x,
(15)
which increases as Fl or x1 increases. If the first feed from the top is controlling,
which increases as F2 increases or x1 decreases. The vapor rate savings are proportional to the feed composition difference between the two feeds for both cases. For difficult separations, the vapor rate savings can be quite large. In a tower with three binary, saturated liquid feeds, a necessary and sufficient condition for Rm,3> R,,z is
> a[F1(1- xi)x3 + Fz(1 - x z ) ~ 3 ] (17) Similarly for R,,z > Rm,l Fzx2(1- xz) + F3x3(l - x 3 ) > cuFlxlx2 (18)
F3(1 - x J ( 1 - XJ
Also, if the first, second, or third feed is controlling, the vapor rate savings are, respectively,
(11)
Binary Mixtures For a binary mixture of saturated liquid feeds, the root of interest in eq 3 is oj
=
a (a - 1)Xj
+1
where x is the mole fraction of the light component in the fth feed. Equations 11 and 12 give
An interesting result of this analysis is the fact that there are vapor rate savings found only when the feeds are arranged in order of decreasing light component composition (xl > x z > X J from the top to the bottom of the column. Equations 19-21 also show the penalty that can be expected for feeding in the wrong order. The feed flow rate has no effect in determining the order of the feeds but only has an effect on the amount of the vapor rate savings. In
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 1841 other words, the feeds should be arranged in order of increasing minimum external reflux ratios (each reflux being calculated in the absence of the other feeds) from the top to the bottom of the column.
Ternary Mixtures For ternary mixtures, we examine double-feed columns with saturated liquid feeds. In eq 11, RO,,fcan be estimated from approximate expressions developed by Glinos and Malone (1984) for saturated liquid feeds or from the expressions in the Appendix of this paper for nonsaturated feeds. For the A/BC sharp split, Glinos and Malone (1984) suggest that
Table I. Rigorous Calculations for Example 1 feeds in correct order 35 10,20 244, 223 21.3, 22.9 60.3 95.2 1.17 1.38
no. of trays feed tray location feed temp, O F feed pressure, psia liquid reflux, lb mol/h vapor a t top, lb mol/h condenser duty, MBtu/h reboiler duty, MBtu/h
premixed feeds 26 13 231 21.8 65.1 100 1.23 1.43
TO VENT
0 GLACIAL ACETIC ACID
I I
+ l C I
A
feeds in incorrect order 35 10, 20 218, 249 21.3, 22.9 72.2 107 1.32 1.52
t
ABSORBER
or unless x F , C is close to Unless aBis almost equal to unity, the minimum reflux ratio increases as (1- x F , C ) / X F , A increases. As discussed before, the feeds should enter from the top of the column in order of increasing minimum external reflux ratios. It is worth noting that when the minimum reflux ratio is the criterion for ordering the feeds, it is possible to have a situation where a feed with the largest light key composition should be the lower feed to the column. A surprising conclusion is that the feeds should not necessarily be fed in order of increasing boiling point (increasing light key composition) from the top of the column: in fact, there is often a penalty in doing so. This can never happen in ideal binary mixtures where the reflux ratio always increases as the boiling point of the feed increases. Levy and Doherty (1986) found one such counterintuitive case by using an independent analysis; the development here gives general criteria for such cases in constant volatility mixtures. A sufficient condition for the second feed from the top feed to control is
where CYB> 8, > 1 is a root of eq 3. Following the same reasoning, for the sharp AB/C split, the feeds must be fed from the top in order of increasing (1- x F e ) / ( l - x F , ~ ) unless , CYBor XFA are very close to unity. A sufficient condition for the uppermost feed to control is
where 1 < 8,
< a B is a root of eq 3.
Example 1 We consider two equimolar feeds, Fl and F2,of n-hexane, n-heptane, and n-nonane with mole fractions of (0.30,0.10, 0.60) and (0.40,0.30, 0.30), respectively. The flow of each feed is 50 lb mol/h. The desired separation is n-hexane from the two heavier components, i.e., a “direct” or A/BC split. The relative volatilities are nearly constant and are 13.76, 5.73, and 1.00, respectively. Contradicting intuition, but according to our criterion, the first feed with the smaller light component composition (and the higher boiling point, e.g., 243 vs. 223 OF, respectively) should be fed closest to the top of the column.
XETDNE
RECYCLED
ACETONE
Figure 2. Flow sheet for the acetic anhydride process.
This is because (1- x F , c ) / x F , A is 1.333 for F, but 1.75 for F2. Under these conditions, the minimum reflux ratio in the double-feed column is 0.96 (the first feed being the controlling one) instead of 1.13 if the feeds are premixed. However, if the feeds are not premixed but are fed in the wrong order, the minimum reflux is 2.02. The minimum vapor rate savings that result are 8.6% when the feeds are fed in the proper order in comparison to the case of premixed feeds. Tray-to-tray distillation calculations using the PROCESS computer-aided design software (Simulation Science Inc., 1985) were done for each of the three cases. The number of theoretical trays and the feed tray locations were estimated by using Fenske’s equation and Gilliland’s and Kirkbride’s correlations (Van Winkle, 1981). The simulations used thermodynamic models based on K values determined on each tray from the Soave-RedlichKwong method. The product specifications were 1% recoveries of the heavy and light keys in the distillate and bottoms. The condenser pressure was set to 20 psia, and the pressure drop per tray was assumed to be 0.15 psia. The results are summarized in Table I. The results confirm our estimates based on the minimum vapor rate, e.g., vapor rate savings of approximately 8.6% with respect to the case of premixed feeds and 16.5% in comparison to the case of feeds that are incorrectly ordered.
Example 2 We consider the separation of acetone from acetic acid in an acetic anhydride process as described in the 1958 AIChE student contest problem; a flow sheet is shown in Figure 2. The streams to be separated in the product column have the flow rates and compositions shown in Table 11. The three streams are binary mixtures, and from eq 17 and 18, we conclude that the first feed is controlling. The minimum reflux ratio is 0.106, which is 41 % lower than the value of 0.229 obtained by premixing the feeds as described in the original case study. We estimate that in a three-feed column, the vapor rate can be reduced by 12%.
1842 Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 Table 11. Feed Rates for the Acetone/Acetic Acid Example. Flow Values in lb mol/h feed 1 2 3
acetone 73 65 31
acetic acid 20 100 81
Perhaps, contradicting intuition, the feed with the lowest flow rate and the largest composition of acetone is the controlling feed. One reason for this is the fact that the separation of the acid from the anhydride is quite easy ( a = 8.3). However, in other cases, when the separation is difficult ( a close to unity) feeds with high flow rates must be expected to be controlling. This can be checked easily if we inspect inequalities (17) and (18) where a appears as a multiple only in the right-hand side. More exact calculations with PROCESS were also performed for this separation to verify our results. The condenser pressure was 14.7 psia, and similar column specifications as in Example 1 were imposed. The vapor fugacities were estimated by the Soave-Redlich-Kwong method and the liquid-phase activity coefficients by the UNIQUAC equation. The vapor savings are 15.9%, and the operating reflux ratios are 0.108 and 0.285 for the case of premixed and multiple-feeds, respectively. Table I11 shows the major simulation results which are in good agreement with the approximate solutions.
Multiple-Feed Columns with Side Streams We consider a column with n feeds and k side streams. Although we still have n reflux ratios, the general equation (10) has to be changed to account for the side-stream flows. The net upward flow of component i below the side stream is Dxd,l - C{lllF,~F,,r + C ~ i l P 1 x qwhere ,, sf is the number of side streams above the fth feed. The reflux ratio for the fth feed can be written as (r",f
f-1
si
/=I
k1
+ 1)(D - CF, - C P J = f-1
si
%(Dxd,t - C F j X F , , i J=1
5
1=1
a1
- Of
+ cp1xP,,1) 1=1 (25)
The reflux ratio can also be written in terms of the internal flows (see Figure 3)
The internal liquid flow to the feed tray is f- 1
s/
j=1
1=1
L f = Lo + CqjF, - CqJ'i
(27)
Thus, the external reflux ratio is
which can be simplified to
' (29) i=l
Table 111. Rigorous Calculations for Example 2 feeds no. of trays feed tray location(s) feed temp, "F feed pressure, psia liquid reflux, lb mol/h vapor a t top, lb mol/h vapor a t bottom, lb mol/h condenser duty, MBtu/h reboiler duty, MBtu/h
multiple feeds 22
4, 9, 16 144, 180, 200 15.2, 15.8, 17 18.2 188
260 2.35 2.66
x
premixed 15 8 I73 15.7 48.2 218 298 2.73 3.06
Ff+l
%+l ps
tl-"&L VN+1
EN
Figure 3. Multiple-feed, multiple-side-stream tower, showing the internal and external flows.
Side streams generally have q1 either 0 or 1, corresponding to a saturated vapor or liquid, respectively. The minimum reflux ratio for the controlling feed cannot be found unless the flows and compositions of the side streams as well as the feeds are known. However, it is not possible to specify the side-stream compositions arbitrarily, since there is a maximum concentration of middle component that can be achieved in a side stream, which is not known a priori. For an n-component mixture with k side streams and specified feed flows and compositions, we have n material balances and k + 2 mole fractions that sum to unity (for the k side streams plus the bottoms and distillate). The unknowns for the design problem are the h + 2 product flow rates and their n(k + 2 ) mole fractions. Therefore, the number of design specifications we have to make are n(h + 2 ) + k + 2(n + k + 2) = n(k + 1). Since there are no limitations on the distillate and bottoms compositions, 2(n - 1) mole fractions may be specified, leaving n(k - 1) + 2 unknowns. Depending on the problem, the rest of the unknowns can be either side-stream purities or side-stream fractional recoveries. No matter what the choice, an estimate of the limiting side-stream compositions is required to ensure feasibility of the column design. However, the estimation of the limiting compositions cannot be done before the determination of flows in the rectifying and stripping sections of the column (i.e., the controlling feed). Thus, after all flows and compositions have been specified, the application of eq 29 will show which feed is controlling, which is vital for the calculation of the limiting compositions. The problem of estimating the minimum reflux ratio for a multiple-feed column with side streams is coupled to the problem of designing the side-stream column, unlike the design of single-feed column with or without side streams.
Ind. Eng. Chem. Res., Vol. 26, NO. 9, 1987 1843 Table IV. ComDarison of the Approximate Design Models with the Stage-to-Stage Process Simulation Results" sequence (a-c) for column (1 or 2) b
a
vapor rate B-component mol fraction
1
2
129 0.990
99 0.990
C -
2 Preliminary Design 147 121 0.990 0.990
d -
e
1
1
1
2
147 0.744
129 0.737
90 0.968
105 0.968
19 8
55 15.41 29 66 106 94 0.98 1.27 1.30
1
PROCESS Simulations no. of trays feed tray location(s) side-stream tray liquid reflux vapor rate a t top vapor rate a t bottom B-component mol fraction condenser duty reboiler duty
26 12
27 13
28 11
26 12
95 135 123
93 123 118
97 167 160
81 121 113
1.62 1.71
1.63 1.66
2.18 2.26
1.44 1.48
55
38 18 129 160 142 0.746 1.91 1.98
55
19 29 102 142 130 0.727 1.70 1.81
48 98 93 1.23 1.31
"Heat duties in MBtu/h, flows in lb mol/h. The top tray is no. 1.
Number of Trays After the controlling feed is found, the number of trays for each column section can be calculated by using the exact Underwood's equations. For example, for any section above the controlling feed tray,
(30)
where & and & are two of the n - 1 roots of Underwood's equation for the actual vapor rate for this section. For every section, we have n - 1 relationships of the form of eq 30. For f feeds and k side streams, the number of column sections is n + k + 1. The complexity of eq 30 can be avoided for a preliminary design if we use Gilliland's correlation for the sections where the vapor flows are near the minimum. For the top and bottom sections of the column, where essentially binary separations are performed, Smoker's equation can also be used.
Design of a Double-Feed Column with a Side Stream We consider a ternary mixture of components A, B, and C. Since we are interested in sharp separations, we estimate the external flows for the case where essentially no C appears in the distillate and no A in the bottoms. There are three alternatives for the side-stream location; it can be taken from above the first feed and below the second or from the section between the two feeds. We examine the case where a saturated liquid side stream is taken above the first feed and the primary separation is the B/C split. The distillate and bottom compositions have no restrictions, so we specify Xd,A and Xb,c. Since Xd,c = Xb,A = 0, two more product specificiations are needed. One of these is the fractional recovery of C in the bottoms; the alternative specification of the composition of A in the top is unsafe since it can conflict with the minimum concentration of A in the side stream which must be below the maximum. The last specification is the recovery or the purity of B in the side stream. These specifications allow us to calculate all of the external flows, the stream compositions, and the minimum reflux ratios from eq 28, for both feeds. The largest R, determines the controlling feed and the actual reflux ratio in the column.
For a feasible design, the mole fraction of A in the side stream found from material balances should be greater than the minimum value (Underwood, 1948)
where @2 is the root of Underwood's eq 32 that lies between CYB and CYC, evaluated at the actual reflux ratio, R ,
Example 3 We consider the separation of a mixture of 100 lb mol/h of n-hexane (A), n-heptane (B), and n-octane (C). The feed is a saturated liquid with mole fractions 0.4, 0.3, and 0.3 and relative volatilities 4.0,2.0, and 1.0. We recover components A and C with a 99% purity, and our objective is to recover a significant amount of relatively pure component B without a large increase in the vapor rate. Some of the possible schemes as shown in Figure 4 are the traditional direct (a) and indirect (b) sequences and columns with a side stream above (c) or below (d) the feed. These schemes can be compared with the prefractionator sidestream configuration (e), proposed by Petlyuk et al. (1965) (case 111). In the latter, a sharp AB/BC separation is performed first and then the bottoms and the distillate are fed to a second column, where B is recovered in a side stream between the two feeds and A and C are the distillate and bottoms products. Table IV shows the results obtained from the approximate design solutions. We see that the double-feed side-stream column with prefractionator can give a high middle component composition with 1770savings in vapor rate over the best simple sequence (direct). The sidestream columns have lower vapor rates but the side-stream compositions are relatively low and may not always be desirable. Although an increase of the vapor rate can produce higher middle component purities, eq 31 is not reliable when the operating reflux increases far beyond the minimum. The calculations were performed for operating reflux 1.2 times the minimum. The main disadvantage of any side-stream column is the limitation on the purity of the B-component product stream, unlike the direct and indirect sequences where there are no constraints. However, for this example, this limiting composition for the prefractionator side-stream
1844 Ind. Eng. Chem. Res., Vol. 26, NO. 9, 1987
_ _ - APPROXIMATE - THEORETICAL
--I
---u :_I-I
-6
-4
20
I
I I
0 50
I
AB/C
075
I
100
150
125
FEED OUALITY
Figure 5. Approximate and theoretical (exact) minimum reflux for a ternary mixture as a function of the feed quality.
Acknowledgment We are grateful to the U.S. Department of Energy, Grant DE-AC02-81ER10938, for financial support.
-C
Figure 4. Various schemes for the separation of ternary mixtures: (a) direct sequence, (b) indirect sequence, (c) column with side stream above the feed, (d) column with side stream below the feed, (e) Petlyuk-type column.
scheme is fairly high, 96.8%. In some designs, this may not be a serious restriction since byproduct streams or recycle streams do not always require high purity. Again, PROCESS simulations were performed for all designs of Figure 4. The condenser pressure for all columns w8s set at 28 psia. The recoveries of the heavy and light keys in the distillate and bottoms were specified as 1?4, and the method of Soave-Redlich-Kwong was used to estimate vapor fugacities and activity coefficients. For the side-stream columns, the recoveries of the lightest (n-hexane) and heaviest (n-octane) component in the side stream were set equal to 1% for the case of side stream below or above the feed, respectively. The multiplefeed/side-stream column with prefractionator had a total vapor rate of 205 lb mol/h, and the purity of n-heptane reached 98%. The approximate design estimations were 195 lb mol/h for the total vapor rate and 96.8% for the purity of the middle component. The total vapor rates and the side-stream purities estimated by the approximate methods (see Table IV) are in good agreement with the rigorous calculations.
Conclusions Simple, analytical expressions for the design of complex columns including multiple-feed/side-stream units are derived. In multiple-feed columns, the controlling feed is the one corresponding to the largest minimum reflux ratio. Simple criteria for determining the controlling feed as well as the ordering of the feeds are presented. The actual reflux ratio depends not only on the feed compositions and relative volatilities but also on the feed flows. Ordering the feeds to the column according to the boiling point is shown to be misleading. Complex column arrangements can reduce the operating costs as much as 30% and lead the process engineer to the new design alternatives.
Nomenclature D = distillate flow rate, mol/h F, = feed flow rate of jth feed from the top, mol/h L, = liquid flow rate to the feed plate of jth feed from the top, mol/h N,= number of theoretical trays for the rectifying section P, = side-stream flow rate of sth side stream from the top, mol/h q = feed quality of jth feed from the top = minimum reflux ratio for the jth feed from the top external) rmJ = minimum reflux ratio for the jth feed above the jth feed tray (internal) S,, = minimum reboil ratio for the jth feed from the top V = vapor rate to the condenser, mol/h V, = vapor internal rate to the feed plate of jth feed from the top, mol/h W = bottoms flow rate, mol/h x , , ~ ,= mole fraction of component i in the jth feed from the top xipa = mole fraction of component i in the sth side stream from the top x,,= ~ mole fraction of component i in the distillate xt,, = mole fraction of component i in the feed to the fth tray z = number of feeds
G
Greek Symbols = relative volatility of component i with respect to the heaviest one 8 = root of eq 3 ( = root of Underwood's equation for the actual vapor rate 4 = root of eq 32 or ratio of the relative volatilities of the key component in the Appendix cy
Appendix A good estimate of R i , f when feed quality is not unity can be accomplished if we expand the reflux ratio of a binary mixture in a Taylor series around q = 1 1).+
+
where = cyLK/aHI< and RMis the reflux ratio at q = 1 (saturated liquid). The approximation gives results in very good agreement with Underwood's method for 0.5 < q
