Ind. Eng. Chem. Res. 1994,33,440-443
440
Combinatorics of Sharp Separation Systems Synthesis: Generating Functions and Search Efficiency Criterion Pascal P. Floquet, Serge A. Domenech, and Luc G. Pibouleau' URA CNRS 192, ENSIGC, 18, Chemin de la Loge 31078,Toulouse Cedex, France This paper gives a generating function of the total number of possible separation sequences when only sharp separators with two or three outputs are used and a generating function of the number of distinct complex separators. These results allow us to define an efficiency criterion for the solution of the optimal separation schemes synthesis problem.
Number of Possible Sharp Separation Sequences: Previous Works The design of sharp separation sequences is one of the most investigated problems in the synthesis of chemical units. I t consists of synthesizing an optimal separation scheme when an n-component mixture has to be separated into pure products. The main assumptions are the following: (1) only sharp separators are used, Le., each component of the feed stream exits in only one output stream of the separator; (2) the components are ranked in any stream; (3)this ranked list of components is invariable; and (4) mixture or division of intermediate streams is prohibited. The number S n of separation sequences may be defined recursively for sequencesinvolvingsimple or complexsharp separators (Shoaei and Sommerfeld, 1986; Wahl and Lien, 1990; Domenech et al., 1991), for example, a distillation column involving a side stream (see the works of Petlyuk et al. (1965), Elaahi and Luyben (19831, or Alatiqui and Luyben (1985) for works using this type of separator in practice). Thompson and King (1972) first presented a closedform expression for S, for sequences involving only simple sharp separators:
s,
(2(n- l))! = n!(n - l)!
Shoaei and Sommerfeld (1986) pointed out that this determination is an application of Catalan numbers. When two- or three-output separators are admitted a closed-form equation can be derived (Floquet et al., 1993):
-
E((n 1)/2)
s, =
Z
(2n - 2 - i)! i!n!(n - 2i - I)!
(2)
where the function E ( x ) represents the bracket function. It can be noted that the first term of the sum (obtained for i = 0) is equivalent to the number of sequences found by the Thompson and King formula and the following terms correspond to the number of sequenceswhere i threeoutput separators (i L 1) can appear. The function E expresses that for the separation of an n-component mixture, the maximum number of three-output columns is equal to E((n - 1)/2). The formulation of a closed-form expression, when the number of outputs for each separator is not specified, is a rather difficult task. For an n-component mixture, it leads to (Floquet et al., 1993)
* To whom all correspondence should be addressed.
I, = (mn-2,mW3,...,m,, mdmo + 2m1 + ... + (i + l)mi + ... + (n - l)m,+2 = n - 1)
Number of Possible Sharp Separation Sequences: Generating Function Use The use of generating functions is an elegant way to derive the number of possible separation schemes. For the Thompson and King formula (11, it is very often described in combinatorics textbooks. It consists of determining the infinite power series expansion of a given function that supplies the coefficients S n of Catalan numbers. For the case of two- or three-output separator sequences, the generating function g(x) is
so+ s,x + s$ + ... + s,xn + ... = 1 + x + ... + S,x" + ...
g(x)=
(4) where SO = 1, S1 = 1,and Sn (n L 2) is given by eq 2. In order to initialize this recursive formula, the values of SO and SI, without any physical sense, were chosen. Then (Wahl and Lien, 19901,g ( x ) is proven to be one of the real solutions of the third degree equation:
+
g3(x) - @ ( x ) + x 1 = 0 (5) The first step of the solution of this classical equation is the elimination of the term -2g2(x): g(x) = f ( x ) + 2/3 So, eq 5 is then equivalent to
(6) (7)
Solving this equation (7) for the case x E I -1, 5/27] (functions f and g are defined for x = 0) leads to three real roots:
4
f3(x) = 5 cos(, -
0888-588519412633-#44O$O4.5Q/O 0 1994 American Chemical Society
F)
Ind. Eng. Chem. Res., Vol. 33, No.2, 1994 441 Table 1. Distinct Beparatorn for an n-Companent Mixture (Two- or Three-Output Separators) no. of distinct TS(n)a + n distinct separators eeparatow TSbh 2 3
4
1 2 2 1 3
AIB A/BC;AB/C A/B;B/C AIBIC A/BCD;AB/CD;ABC/D A/BC;AB/C;B/CDBC/D A/BB/C;C/D A/B/CD;A/BC/D;AB/C/D A/B/C;B/C/D
by n-1
TS(n), = c i ( n - i)
1
n22
1-1
TS(n),=
5
"-2i(n - i)(n - i - 1)
n23
PI
4
3 3 2
15
with Then From eqs 4 and 6 it follows that f(0) = 1/3 and (df/dx)(O) = 1, and therefore solutions f&) and f&) can be eliminated. Finally, the generating function g ( x ) is g ( x ) = cos cp
+ -23
n-0-1)
i(n - i)!
TS(n), =
n l r
(13)
and (9)
n
TS(n) = &TS(n),
with
r=
and x
E 3 -1,5/27 [ (nx;l;blank
Number of Distinct Sharp Separators: Generating Function Use It can be seen, for example, that the 4-component synthesis problem has 5 different sequences of simple separators. Each sequence involves 3 separators, giving 15 separators all in all. In fact, there are only 10 distinct simple separators (seeTable 1).In the same manner, there are 10 sequences of 2- or 3-output separators for the 4-component problem, but only 15distinct separators. The number of distinct separators, as noted by Wahl and Lien (1990)TS(n), for an n-componentmixture to be separated with F output separators, is important in practice because it gives a lower bound on the total computational effort. The values of TS(n), and TS(n) (TS(n) represents the total number of distinct separators, having from 2 to n outputs, for separating an n-component mixture) given by Wahl and Lien (1990) are the following:
These values constitute the lower part (from the 4th column) of the Pascal's triangle, as it is shown in Figure 1. The use of generating functions to derive the above formulas (13 and 14) is now presented. Let iz(x) the generating function of the coefficient TS(n)z: i2(x)= x2( 1+ 4x
+ 10x2 + 2oX3+ ... + (:")xn+
...) (15)
that is to say in a developed form OD
i2(x) = x 2 z -
(i + 3)! . 3!i!
2'
and dividing by 3! it becomes (18)
and finally n+l TS(n), = (4
)=
n4-2n3-n2+2n 24
i2(x)= x2(1- x14 =
X2 (1x)4
A generalization of this result to separators with r output streams leads to i,(x) = x3(1 -xl4 =
forr = 3
(20)
x)6'
and i,(x) = ~ ' ( 1 x)-'-~ = From Table 1,it can be proven recursively (see Appendix) that the values of TS(n), and TS(n) can also be expressed
-(1x-3
Xr
(1- X)*2'
*
for the general case
Thus, the generating function of TS(n) is given by
(21)
442 Ind. Eng. Chem.
Vol. 33, No. 2, 1994 TSZlnl TSXnI
1
n=o n=1 "12
1 1
n=3 " 1 4
n=5 "=6 n=7
1
n=9
1
"=lo n.11 n=12 n=13
1
1 2 3 4 5 6 7 8 9
1
10 11 12 13 14
1 3 6 10 15 21 28 38 45 55
1 8 42 00 210 488 088 1081 4017 8100 16278
86
18 91
Figure 1. Computation of TS(n) Table 2. Number of Distinct Sepmtora for am a-Comwnent Mixture n 2 3 4 5 6 1 8 9 4 10 20 35 56 84 TS(n)z 1 rz 100% 100% 61% 36% 11% 1% 3% 1 5 15 35 IO 126 TSWa 0 r3 100% 100% 60% 28% 11% 4% 1.3% 5 16 42 99 219 466 TS(n) 1 r 100% 100% 61% 30% 13% 5% 2%
1
0 165 120 0.4% 1% 210 330 0.4% 0.1% 968 1981 0.6% 0.2%
(23)
Nomenclature g ( x ) = generating function of number of sequences S . i,(x) = generating function of number of distinct separators TS(n), I . = set of admissible structures of sharp separators m = total number of separators in a sequence = E m E p t k ) r n k = number of separatorswith one input and (k+ 2) outputs n = number of components to be separated ri = ratio of distinct separators to the total number of separators of all sequences (separator with one input and less or equal to i outputs) S . = number of separation sequencea of an n-component mixture TS(n), = number of distinct separatorswith r output streams, for an n-component separation TS(n) = total number of distinct separators for sequencea of separation of an n- component mixture Appendix: Derivation of the Equivalence of Relations (10 a n d 12) This equivalence is derived recursively. For n = 2
Conclusion Table 2 shows the values of TS(n)z, TS(n)3, and TS(n) versus n and the ratio of distinct separators to the total number of separators of allsequences, in the case of simple separators (one input, two outputs) or complex separators (one input and more than two outputs). These ratios can be computed according to relations 26-28 TS(n), r, = ( n US,
-
r3 =
value of these ratios, it can be considered like quite inefficient. So, these ratioscan beinterpretedasefficiency criteria for combinatorial optimization techniques.
-- n!(n - 2)!TS(n), (2(n - l))!
TS(2), =
(i)
= 1,from (IO); TS(2), = 1,from (12)
If, for n 2 2, relations 10 and 12 are equivalent
Then (26)
n
n
r-1
z-1
TS(n+l), = z i ( n + 1- i) = z [ i ( n- i) + il
TS(n), + TS(n), E(W)/21(2n- 2 - i)!(rn- i - 1)
(27)
+ p1( n + l ) -- ( n- l)n(n + 1) 3n(n+ 1) = TS(n),
+
6
- n(n + l)(n + 2)
6
6
with the same definition of rn and I.. The main feature of the mathematical developments presentedinthispaperliesinthedefinitionoftheseratios. Indeed, they constitute a lower bound on the number of designed separators, for solvingan n-component problem. If a given algorithm (Branch and Bound, Simulated Annealing, etc.) screens the solution tree more than the
that proves the equivalence for n + 1. The derivation of the other relations (for TS(n)3, ...,TS(n),) is made in the same way.
Literature Cited Alatiqui. J. M.;Luyben. W.L.Altemtivediatillatioueonfiguratioas for =parating ternary mixtures with small concentration on intermediate in the feed.Id. Eng. Chem. Process Des. Deu. 1986, 24 (2), 500-511.
Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994 443 Domenech, S.; Pibouleau, L.; Floquet, P. Dbnombrement de cascades de colonnes de rectification complexes. Chem. Eng. J. 1991,45, 149-160. Elaahi, A.;Luyben, W. L. Alternative distillation configurations for energy conservation in four-component separations. Ind. Eng. Chem. Process Des. Dev. 1983,22 (l),80-87. Floquet, P.; Pibouleau, L.; Domenech, S. Agencement de colonnes de rectification complexes. Chem. Eng. J. 1991,47,119-135. Floquet, P.; Domenech, S.; Pibouleau, L.; Aly, S. M. Some complements in combinatorics of sharp separation systems synthesis. AZChE J. 1993,39 (6),976981. Petlyuk, F. B.;Platonov, V. M.; Slavinskii,D. M.Thermodynamically optimal method for separating multicomponent mixtures. Znt. Chem. Eng. 1965,5 (3),555-564.
Shoaei, M.; Sommerfeld,J. T. Catalan numbers in process synthesis. AIChE J. 1986,32 (ll), 1931-1938. Thompson, R.W.;King, C. J. Systematic synthesis of separation schemes. AIChE J. 1972,8,941-953. Wahl, P. E.;Lien, K. M. Combinatorial aapecta of sharp eplit separation systems synthesis. AIChE J. 1990,36(lo),1601-1613. Received for review October 1,1993 Accepted December 1, 1993. e Abstract published in Advance ACS Abstracts, February
1994.
1,
