Calculation of Minimum Reflux for Distillation Columns with Multiple Feeds Francisco J. Barnes, Donald N. Hanson,' and C. Judson King Department of Chemical Engineering, University of California, Berkeley, Calif. 94780
Underwood's minimum reflux equations for distillation columns are extended to columns with two or more feeds. A Fortran computer program for calculation is available from the authors. The program is general for 15 feeds and 220 components.
A
particularly convenient method for the calculation of minimum reflux for distillation columns was described by Underwood (1946,1948) and elaborated by Shiras e t al. (1950). The method requires assumption of coiistant Eelative volatilities and constant flows for what may be substantial portions of the column, but despite the uncertainty introduced by these assumptions, the method is accurate enough for extensive use. Other methods, notably iteration methods, can be made as accurate as the basic data permit but at the expense of a considerably more complicated calculation. However, since it is seldom necessary to know minimum reflux with great accuracy, there has been little incentive to develop these alternate procedures. Underwood's method wah originally worked out for columns with a single feed, and its application to columns with more than one feed has not been shown. The authors recentlyneeded a calculation method for columns with multiple feeds and extended Underwood's equatioiis to these caies. I n this extension the method still retains sufficient simplicity to be most useful. Even a? one increases the number of feeds and the complexity mounts, it is probably still preferable to a solution by iteration techniques. Only three basic equations are necessary: First, a n equation relating the vapor flow to various parameters, +, for a particular section
ffi
 $k
fff

$7
where (xi), is the mole fraction of component i a t the top of the section, ( Z J Z is the mole fraction of component i at the bottom of the section, $ j and $k are two parameters for the section obtained from Equation 1, and n is the number of equilibrium stages in the section. (In this work n = 00 .) Single Feed Columns
To demonstrate use of the equations, i t would be desirable to review briefly the calculation of minimum reflux for a column with a single feed. To facilitate this examination, consider a specific example of a feed of five components, labeled alphabetically in order of decreasing volatility a s A , B, C, D , and E . A minimum reflux problem is defined by specifying the separation on two of the components, the light and heavy key components, and by specifying infinite stages in each section. If the keys are taken here as B and D ,A is then a light diluent, C i s a sandwich component, and E is a heavy diluent. Assume that components A and E both distributethat is, they appear in both products. Equation 1 then yields five values of $ for the rectifying section and five values of for the stripping section which can be denoted by their magnitude in relationship to the relative volatilities of the components as follows:
+
where V is the vapor flow rate in the section, CY% is the relative volatility of a component, P is the total net upward product in the section, and (xOpis the mole fraction of component i in the net upward product. The summation is taken across all components. Second, a n equation for the calculation of all possible parameters, a, which satisfy Equation 1 for twocolumn sections which meet at a feed to the column
,F(z i) F V F = Z p ff
fft

0
(2)
where V Fis the change in vapor flow owing to introduction of feed, F(zi)Fis the amount of component i in the feed, and 0 is termed a common root of the two sections. Third, a n equation which links liquid compositions at the two ends of each column section To whom correspondence should be addressed. 136 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
($RB)AE
(6SS)AI
($RS)EC
(4SS)AE
($RS)CD
(6SS)BC
($RS)DE
('#'SS)CD
(@RS)E
($SS)DE
where ( + R ~ ) C D lies in value between LYC and LYD, and (+RB)Elies below CYE, etc. If the values of from the rectifying section are inserted in pairs into Equation 3, together with ( z , ) ~= (xJd (the mole fraction in the top product) and ( x i ) $ = ( x t ) f (the mole fraction on the feed plate) , and $? is less than $k,
+
fft
 +&
(4)
If the values of + from the stripping section are similarly used, together with (zJl = (xJ0 (the mole fraction in the
bottom product) and
=
(z*)f,and $?is less than $k,
(+IS)
AB
E
fft(Zt)f = fft
 $9
BD1
(4) *
BD2
The list of $ values which satisfy Equation 4 is thus
DE Two roots are now found between the avalues of the components where the sign of P(z,), changes from to . The labeling convention has been adopted that ( $ I S ) B D ~< ($lS)BDl. If these roots are inserted in Equation 3 in adjacent pairs,
+
However, t'here can be only four roots (or 4 values) to Equation 4,and apparently, each of the pairs above must be equal. Theqe common roots of the two sections must be roots of Equation 2e.g., ( : Q R S ) B C = ( $ S S ) B C = (@F)Bcand can easily be found from that equation. They can then be inserted in Equation 1 written for one of the sections, yielding jupt ) ~VRs. , enough equatioiis to calculate d ( r A ) d ,d ( ~ )d ~( ,~ ~and V R Sis t'he minimum vapor flow for the separation specified on components B and .D since it corresponds to infiiiit'e stages in both sections. Often the values of 1d(ZA)d and d ( ~are~ such ) ~a s to be impossible, d(za)d > F ( z ~and ) ~ d ( x E ) d < 0, indicating that 9 and E do not distribute. The common roots associated with these components cannot be used, and t'he amount's of these components in the distillate must then be set a t the amount in the feed or zero. This contingency can be shown as follows:
where (xJIl and (zJf2are mole fractions of a component on the upper and lower feed plates, respectively. These solutions of Equation 3 must be considered along with similar solutions obtained with values of ~ R and S $85 to determine where common roots lie. I n the following listing, all of this important information has been condensed. P(XJP
9RS
91s
9SS

AB BD DE E
AB BDl B0 2 DE
A+ AB BD DE
+ +
A B (lk) D (hk) E
E ffi(Xdf1= 0 [31 ffi

2ffi(Zi)p = 0 [31 ffi 4
$
($s~)AB
($RS)AB
(4R S ) A B
( $ s s ) A ~[if
(+RS)BC
( 4 S S ) BC
($RS)CD
A distributes]
($RS)BD ($RS)DE
($SS)CD
[if E distributes]
(~RS)DE
($SS)DE
($IS)BDP
Only if A distributes is the common root ( @ p ) a B valid for use in Equation 1 , and oiily if D distributes is the common root ( @ p ) D E valid. However, this is not a problem since there are always just enough varlid common roots to determine the unknown distributions and the flow. Two Feed Columns
($SS)BD

(if E distributes)
+
($SS)DE
(4IS)BDl ($IS)BDZ
First, the signs of P(z,), are shown, together with all of the roots which are obtained from Equation 1 for the threecolumn sections. The two equations which result from Equation 3,
E a t ( z t ) f l
Solutions of columns with two feeds use the same equations and follow the same general procedures. HoJvever, since a n intermediate section of infinite plateq has been added to the column, and the common roots can come from either feed equation, the solution is more complicated. To illustrate the method, consider a relatively simple case comprised of the previous example without the sandwich component. Components A, B , D , E then constitute both feeds. The upper feed is labeled F 1 and the lower F 2 . The separation of the key components B and D is again specified, thus setting all flows of the key components; in most cases the net upward flow of B in the intermediate section would be and the net upward flow of D would b e  . It would be logical then to assume that P ( x A ) , would be and that P ( z E ) , would be . If the values of P i x , ) , have these signs, the roots calculated for the intermediate section from Equation 1 will be
+,
($IS) DE
(if A distributes)
fft
6
=
0 and 2 ff&t)P = ai

$
are shown next, followed by the number of possible roots for each equation in each case, here, three. Below each equation are listed the roots which can or must satisfy the equation. Assuming that A and E both distribute, there are three roots, $ R S , which must satisfy the equation for the upper feed plate and three roots, $ss, which must satisfy the equation for the lower feed plate. I n addition the roots for the intermediate section which can satisfy either equation are shown. The arrows indicate that the lefthand root must satisfy the upper feed plate equation, or the righthand root must satisfy the lower feed plate equation. Various choices of roots from the intermediate section are apparently possible, but if all of these choices are examined, the only possible ones are those shown by the horizontal lines under certain of the roots. Thus, must satisfy the lower feed plate equation, and ( + i S ) D E the upper feed plate equation. Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
137
However, with the two (+/,y)BD roots which result from the sign change in P ( Z ? ) two ~ , possibilities arise. EiOher ( + I S ) B D 2 must satisfy the upper feed plate equation, or (41s)BDi must satisfy the lower feed plate equation. There are t'wo possible set's of common r o o k from Equation 2 which could give a solution for minimum reflux.
{ if::;:^ 1 { :fl:;:~1
+
+
or
(@F1) DE
+
is present, the to  change can take place between the light key and the sandwich component or between the sandwich and the heavy key component, apparently yielding four possible solutions. However, two of these will be the same; hence, only three possible solutions result. I n general a n additional solution can be found for each possible additional position of the to  change, and the change can occur between any pair of adjacent components from the light key to the heavy key if P ( Z ~is ) ~and P ( z ~ Ais).~ To illustrate these possibilities, consider the previous example with the sandwich component present. First, assume that the sign change occurs between components B and C; the resulting set of roots and their possible use are as follows:
( a F 1 ) DE
Either of these sets of roots can be inserted in Equation 1 written for the intermediate section (since they all apply to the intermediate section) t'o calculate (Vzs)minand P ( Z A ) P and P ( Z ~ If ) ~A. and E do not distribute, the common roots ( @ = ~ ) A Band ( Q i F 1 ) D X are not' valid for insertion in Equation 1 , and either ( Q i F i ) B D or ( @ F z ) B D is used to calculate (Vzs)min alone. Thus, two answers to t'he problem are obtained, and i t is necessary to determine which is correct. The same result is wellknown in solving minimum reflux for twofeed binary distillation problems. If a McCabeThiele diagram is used, clearly, the correct answer is the higher flow of the two calculated. The lower flow always requires that const'ruction of stages take place outside the equilibrium curve in a way not physically possible. The question can be completely resolved in each case by determining the feed plate compositions and calculating from each of them in both directions to the nearest point of infinitude. If this is done, only one solution matches; all other solutions proceed to a match through regions of negabive concentrat'ions for one or more components. These calculations have been done for 16 widely different examples calculated by the authors, and in every case the solution which requires the highest flows is the correct one. Several other points about the solution should be noted. In the preceding problem the sign of P ( Z ; )for ~ the light diluent was assumed to b e and t'he sign of p ( ~ ;for) the ~ heavy diluent' was assumed to , so that the signs of P ( z ; ) P were here +  . If the list of diluents is expanded and the resulting roots are considered, a general rule can b e formulated as follows: If the components are listed in decreasing order of volatility, there will be one sign change from to in the list of P ( z J Pvalues for the components. All signs must above the sign change, and all signs must be  below be the sign change. Any other assumptions will lead to roots from the intermediate section which cannot yield a solution. Another general point is that a valid common 9 root will be found from Equation 2 written for the lower feed, F 2 , between the avalues of all pairs of adjacent distributing components whose P ( z J P values are A valid common a root will be found from Equation 2 written for the upper feed, F1, between bhe avalues of all pairs of adjacent distributing components whose P(.Jp values are . A valid common Qi root lying between the Uvalues of the adjacent' components whose P ( z i ) p values are and  can come from either F1 or F2. Thus, there are always two apparent solutions as noted previously.
The common
P(X,)P
9RS
dzs
9ss

AB BC CD DE E
AB BCl BC2 CD DE
A+ AB BC CD DE
+ +
A B (lk) C D (hk) E
Qi
value sets which must be investigated are
+,
+
+
+
+.
+
Sandwich Components
The presence of sandwich components increases the amount of calculation necessary because each sandwich component adds one additional possible solution, When no components exist between the keys in volatility, the + to  change in the signs of P ( z , ) takes ~ place between the key components, and two possible solutions are found. If one sandwich component 138 Ind.
Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
Second, assume that the sign change occurs between components C and D. The results are
A B (lk) C D (hk) E
P(xi)P
4RS
91s
9ss

AB BC CD DE E
AB BC CD1 CD2 DE
A+ AB BC CD DE
++ +
The common @ value sets which must be investigated are (@FZ)AB
(@Fz)BC (@FP)CD
( @ F J DE
i@SS)CD
Adding these sets of common @valuesto those for the other shows three sets which must' be invest'igated. l f t e r calculation of the possible solutions, it may be found ~ result from the calculation are t h a t the signs of P ( S , ) ,which different from the signs which were assumed in choosing the common @ values to be used. Such a solution is of course wrong if associated with the assumed set of signs. However, for each sandwich component present in the system, there are two sets of signs which yield the same answer, and the solution may well correspond to the other set of signs. I n some cases with certain positions of the to  change in the list of components, the common root's lying between the avalues of t'he two componeiits at which the change has been assumed to occur are in the wrong ratio. T h a t is, the common root from F l is larger than the commoii root' from F2. Such a situation cannot be solved by these equations, as is evident from Equat'ion 3; the smaller root must satisfy
DE
+
ffc
The resulting sets of common roots to be investigated are
9
and
and hence must be a common root from F1. If the ratio of the two roots is in the wrong direction at a particular to change position, the problem cannot be solved a t that position of the sign change. If this is true at all possible positions of the sign change, the feeds are inverted from their optimum order. If it is true a t even one position of the sign change, probably both feed orders should be tried to det'ermiiie which is optimum.
and (@F~)cD
(@FI)CD
(@FI)CD
+
+
If the to  change is placed between C and D in P l ( x , ) p ~ and between B and C in P 2 ( z J P 2and , then the to  change is placed between C and D in both intermediate sections, the resulting new sets of common roots are (@F3)AB
(@F3) A B
(@FZ)BC
( @ F 3 ) BC
(@FZ)CD
Multiple Feeds
(@F1) D E
Problems with multiple feeds can be solved by the application of the same principles. Consider t'he same example but with three feeds. There will now be two intermediate sections, the upper labeled a s I,S1 and the lower as ZS2. Whereas with two feeds all possible positions of the to  change in P ( z J P had to be considered, now all possible posit'ions of the to change must be considered in the lists of bot'h P 1 ( z J F 1and P ~ ( Z * The ) ~ Zrelative . positions of blie t'wo changes are not completely independent; the to  change must, occur a t the same level in the list or higher in the list for the lower section. Otherwise, one finds the net flow of a component such that, it is being brought from two direct'ions to a feed plate, a clear impossibilit'y. ll'ith this restriction the basic information can be shown for a n example with three feeds as six possible solutions.
+
+
+
+
and
(@FB)CD
(@ F J
DE
(@F3)AB
and
( @ F 3 ) BC (@FZ)CD ( @ F I )D E
These six sets of common roots constitute all possible sets. The use of one of the sets will yield the correct solution to the problem. The possible sets of common roots can be easily constructed from the lists of signs for the various intermediate sections. Three rules are needed. Consider two adjacent' component's, j and k , in the list of P ( X ~for) a~ part'icular intermediate section.
If P ( z j ) , and ( P z ~are ) ~both +, the common root @jk cannot b e obt'ained from the feed at the top of the section. If P ( z J Pand P ( z J P are both , the common root @jk cannot be obtained from the feed a t the bottom of t'he section. If P ( z J Pand P ( z l ~are ) ~ and , no restriction is plaeed on obtaining @jk from the feeds at the ends of the section.
+
Pl(X,)Pl
p2(Xi)FZ
A B (lk) C
+ +
+ +
D (hk) E


@RS
@IS1
@ZS2
@SS
AB BC CD DE E
AB BC1 BC2 CD DE
AB BCl BC2 CD DE
A+ AB BC CD DE
T o illustrate the use of the rules, consider the previous example. Three combined sign change positions are possible and are listed below together with the resulting allowed common roots. Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 1,
1972 139
Allowed common roots
from F 3 ( @ ) B c from F 1 , F2, or F 3 ( @ ) c D from F1 ( @ ) D E from F1 (@)AB
from F 3 from F2 or F 3 ( @ ) c D from F 1 or F2 ( @ ) D E from F1 (@)AB
(@)Bc
(@)AB
from F 3
( @ ) B c from F 3 (@)cD (@)DE
from F 1 , F2, or F 3 from F1
The rules for implementation of the solution have been developed to utilize the signs for the various lists of P ( z J p . Note that the sign change can be placed in a particular position in any list between P ( z ~arid ) ~P ( z J P ,only if the common root @jk from the feed above the intermediate section is less than the common root a j k from the feed below the section. Missing Components
A small degree of complication is added to the solution when components are missing from a feed. If a component does not appear in a feed, one of the common roots obtained from the feed and lying immediately above or below the relative volatility of the missing component becomes equal to the volatility of the component. If this root is then used in Equation 3, the equation can be satisfied by a zero concentration of the missing component on the feed plate to which the feed comes. This zero concentration can of course be reached only if the net flow of the component is zero in the column sections both above and below the feed. When the calculation requires the use of a root equal to the volatility of a component, usage of the root is simply replaced by the knowledge that the net flow of the component is zero in the appropriate column sections. The whole problem can be avoided in practical calculations by simply inserting a small amount of each missing component into each feed in which it does not appear. The value of minimum reflux calculated will not be significantly affected by these small amounts of added component feeds, nor will the distribution of other components. Only the distributions of the added components themselves will show a small error. It is commonly found that components which are missing from a feed do not distribute. Thus, a light diluent absent from the lowest feed in the column cannot distribute, nor can a heavy diluent absent from the uppermost feed. It is foreign to the experience obtained from calculations with
140 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
single feed columns but also common to find that sandwich components do not distribute if not appearing in all feeds. Equally surprising perhaps is that diluents which one would expect not to distribute often exhibit a broad distribution. If the column has only a single feed, the diluents must be quite close in volatility to the nearest key component to distribute at all. With two feeds and the light key component as an example, if the net upward flow of the light key in the intermediate section is small, light diluents fed in the lower feed can attain high degrees of distribution between products, even when widely difierent in volatility from the key component. I n the discussion throughout this paper, it has been assumed that the net upflow of the light key component in the intermediate sections is plus, and the net upflow of the heavy key component is minus. If both were minus in a particular intermediate section, then all sections below are really a mixer for the key components. Or if both were plus, then the sections above constitute a mixer for the keys. A description of separations of the key components which do not yield net upflows of the light key and net downflows of the heavy key in the intermediate sections thus appears to represent a faulty description of the problem, and the vast majority of problems would be set so that the flows of the keys had the proper signs. However, it might be desirable under certain conditions, for example, when a feed has a large amount of diluent and a small amount of the key components, to introduce the feed into the same column to separate the diluent from the keys rather than design a separate column for their separation. Such problems can be calculated. The plus to minus change in P ( Z , )for ~ the section is no longer restricted to the key components and the sandwich components but will occur somewhere in the list of diluents. Because of the complexity of the calculations, the authors have prepared a Fortran computer program which is general for 15 feeds and 220 components. Some typical calculation times on a CDC 6400 are: 0.49 sec for a 7component1 2feed system with no sandwich components 1.96 sec for a 20component, 2feed system with 1 sandwich component 1.97 sec for a 4component, &feed system with 2 sandwich components The program is available from the authors on request. literature Cited
Shims, R. N., Gibson, C. H., Hanson, D. N., Znd. Eng. Chem., 42, 871 (1950):
Underwood, A. J. V., Chem. Eng. Progr., 44, 603 (1948). Underwood, A. J. V., J.Znst. Petrol. (London), 32, 614 (1946). RECEIVED for review April 9, 1971 ACCEPTED August 16, 1971 The authors thank the Instituto Nacional, de la Investigacion Cientifica, Mexico, for support of one of their members.