Computers chem. Engng Vol.20, Suppl.,pp. $201$205, 1996
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Synthesis of Mass Exchange Network Using Process Graph Theory Seungkwon Lee and Sunwon Park Department of Chemical Engineering Korea Advanced Institute of Science and Technology 3731 kusongdong, yusonggu, Tajeon, Korea
A B S T R A C T The objective of this paper is to propose a method for synthesizing the optimal mass exchange network (MEN). In order to synthesize optimal MEN, it is necessary to determine the network structure and its operating conditions. The material and operating unit nodes that are essential to use the Process graph theory (Friedler et al., 1992) are defined. The maximal structure of MEN synthesis problem can be easily obtained using the Pgraph theory. Furthermore, only feasible structures can be extracted from the maximal structure, the size of the search space is extremely reduced. The feasible structures are formulated as nonlinear programming problems (NLP) and their operating conditions are determined by optimization of NLP.
INTRODUCTION
must be removed, y[, and an upper bound for this, yU respectively (imposed either by environmental regulation or economical considerations).
The synthesis of MEN is addressed in this paper, where a number of rich streams are integrated with lean streams in order to meet process specifications (eg. environmental regulations) on their final compositions. E1Halwagi et al (1989) first proposed the problem of synthesizing mass exchange networks (MEN) using Pinch technology principles. The major limitation of the Pinch based method for the synthesis of MEN is that cost parameters are not simultaneously optimized and the network optimality refers only to operating cost. Furthermore no systematic method is provided for the derivation of the network config
* A set, L e a n = {JlJ, ..., N L } , of lean streams (or Mass Separating Agents), which may be process streams or auxiliary external lean streams, their mass flow rates L~, their initial compositions, x~ and an upper bound to their outlet compositions, x~. Also, an upper bound to the mass flow rate that can be used in the network, L y and a cost associated with the consumption of each lean streams, CLj.
uration. Papalexandri et al. (1994) formulated the MEN synthesis problem as a mixed integer nonlinear programming problem (MINLP) using hyperstructure representation. Because they formulated the piping segments as well as mass exchangers as binary variables, the number of binary variables is very large (see Example 1). In this work, two step approach is developed to synthesis the optimal MEN: (1) Evaluation of all leasible MEN structures; (2) Determination of operating conditions. In the first step, combinatorial aspect of MEN synthesis is handled by Pgraph theory so that only feasible MEN structures can be generated. The feasible MEN structures determined by Pgraph are formulated as nonlinear programming problems in the second step. PROBLEM
Equilibrium relations between rich stream i * and lean stream j: Y~ = ¢ i j (x~)
* A minimum composition approach e~j to ensure feasible mass transfer in each mass exchanger operating unit node (analogous to the AT, am in the HENS problem). This ¢ does not in any case prespecify the lean stream consumption. * The types of mass exchanger operating unit (eg. perforated plate, packed tower). • The objective is to synthesize an optimal MEN, that can satisfy the specifications for the rich and lean streams.
STATEMENTS
The following assumptions are made:
The synthesis of mass exchange network is to determine the optimum network structure of the MEN.
1. The mass flow rate of each stream remains constant throughout the network.
Given informations are:
2. The equilibrium of considered components does not depend on the other solutes.
• A set, R i c h = { i l i , ..., N R } , of rich streams, their mass flow rates Gi, their initial compositions, with respect to a component that rm zo,m,L)H
(1)
3. The mass exchangers are considered as the countercurrent type. $201
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European Symposiumon Computer Aided Process Engineering6.Part A
4. No mass exchange between richrich and leanlean streams is allowed, 5. Temperature and pressure are constant throughout the network for each stream, so that one equilibrium relation can be applied.
With the above sets, MEN synthesis problem is represented as a Pgraph (see Fig. 1). If we input M, R, P, and O to the Maximal Structure Generation (MSG), we can simply get the maximal structure p(P, R, O) (Friedler et al., 1993). From #(P, R, O), all of the feasible structures that satisfy the Pgraph axioms can be pro
MEN
SYNTHESIS
PROCE
DURE In the proposed method, a two step procedure is used to synthesize the optimal MEN as follows: 1. E v a l u a t i o n of all feasible M E N s t r u c t u r e s : Rich and lean streams informations are converted to the material sets of the Pgraph theory, mass exchange units informations are converted to the operating unit set. Maximal structure which is the union of all feasible network structures are generated, and every feasible network structure is generated from it. 2. D e t e r m i n a t i o n of o p e r a t i n g c o n d i t i o n s : An NLP is formulated for each network structure generated in step 1. Its optimization provides the operating conditions of the structure,
E v a l u a t i o n of all f e a s i b l e M E N s t r u c tures In order to use the Pgraph theory, we have to define the material set M, product set P, raw material set R, and operating unit set O. Rich streams are converted to rich raw set (RR) and rich product set ( R P ) . Lean streams are converted to lean raw set (LR) and lean product set (LP). Mass exchangers are converted to operating unit set (O). These sets are defined as follows:
M
=
RRURPNLRALP
(2)
R P 0
= = =
RRULR RP {O~jli e Rich and j E Lean}
(3) (4) (5)
where
RR RP LR
= = =
{RR41i e Rich} {RPiti E Rich} { L R j l j • Rich}
LP Oij
= =
{LPjl j • Lean} ( { R R i , L R 3 } , {RPi, L P j } )
(6) (7)
duced.
If a solution structure is (m,6[m]) generated by Solution Structure Generation (SSG) ( Friedler et al., 1995), we can define the following material set M' and operating unit set O' : O'
=
op(~[m])
(11)
M'
=
m t3 mat(O')
(12)
where ~[m] is the decision mapping of m , op(d) is the set of operating unit nodes involved in the decision mapping d , and mat(o) is the set of material nodes involved in the operating unit set o.
D e t e r m i n a t i o n of o p e r a t i n g conditions For each feasible MEN, an optimization problem can be formulated as a nonlinear programming problem where the total annualized cost (TAC), expressed in terms of operating and investment cost, is minimized. The variables of rich stream i and lean stream j are shown in Figs. 2 and 3. The model consists of • Variables:
 gi~: Flow of rich stream i from the initial splitter of the stream to the mixer preceding the mass exchanger O~j. gijME" Flow of rich stream i through the O~j. _
 a°~ ~,~ t'• Flow of rich stream i from the splitter after the Oij to the final mixer of the stream. bypass  gij,j : Bypass flow of rich stream i from Oij, to the Oij.  l)~: Flow of lean stream j from the initial splitter of the stream to the mixer preceding the Oij. ME.. Flow of lean stream j through the lji

(8) (9) (10)
O~j is a mass exchanger where rich stream i and lean stream j meet. It has two input material nodes (RR~, L R j ) and two output material nodes (RP~, LI='j). L P is excluded from product set, because it is not necessary to use all lean streams,
Oij.
 lj °.~'t" Flow of lean stream j from the $ • splitter after the Oij to the final mixer of the stream. _ ibypas8 ' ~j~,i • Bypass flow of lean stream j from Oi,j to the Oij.  MTij: Mass exchanged at the unit corresponding to the O~j.
European Symposiumon Computer Aided Process Engineering5.Part A
lJMiE = l~n +
in. Composition of rich s t r e a m / p r i o r  Yij. to the Oij. _ yOy: Composition of rich stream i after the Oij. y~: Final composition of rich stream i i.e. composition of rich product node
tbYPass "ji'i
overall mass balances at the splitter after each exchanger for each stream
gijME
RPi. in. Composition of lean stream j prior  xji. to the 
E Oi,jEO'


$203
=
gO?t " +
bypass E gijj' o,j, eo'
=
out.. Composition of lean stream j after xj~ the Oij.
°t,,"bYPass o~,jeo'

 x ~? Final composition of lean stream j J" i.e. composition of lean product node
L P t.  Nst~t: Number of theoretical plates in perforated plate column Oit, when Oij
mass balances for transferable component at the mixer preceding each exchanger for each stream
gitMEyijin = gitinYir+
E go'tbyPaSSYiJ'°ut o,j, ~o'
is a perforated plate column. 
,ME in = ltiin XJr ~ tit X3i
Hit:Height in packed column which is determined by the number of equilibrium stages and the equivalent height of each equilibrium stage. Oij, when Oit is a perforated plate column,
E
MTij = gijME,(Yijin  yOyt)
in gij
MTij = t ,MEt out Xt~) in t i ~Xti
o,,eo'
 t h e r m o d y n a m i c feasibility constraints for each each exchanger
In the above equation, giji,~ is added only if Oit is included in the set O'.
y~j~t > mij (x~n + ~) + bij
• Overall mass balances and bounds at the lean raw material nodes when L R j E M':
Lj =
E l}7' o,jeo,
Yijm _> m.,J.(x ~ ti°ut + e) + bit
Lj <_ L U
• Objective function (total annualized cost)= operating cost + fixed cost:
• Mass balances for the transferable component and bounds at the rich product nodes when RP~ C M':
GiYP =
gitout Yitout , E o,~ eo'
TAC =
Yip < yU
• Mass balances for the transferable component and bounds at the lean product nodes when LPj e M':
EXAMPLE
tji Xji ~ X j o,~eo'
• Mass balances and constraints at the operating unit nodes (mass exchanger) when
Oij E &: 
overall mass balances at the mixer preceding each exchanger for each stream
gijME
:
gijin +
E 0 o, EO'
gij'jbYp~ss
E CLtLj + E ~(00) LRjeM' OijeO'
where ~(Oit) = ¢ l ( N s t o ) when Oit is a perforated column and O(Oij) = ~2 (Hit) when Oi3 is a packed column.
e r y
~
tti, i.bypassyti,out
 mass balances for transferable component at each exchanger
• Overall mass balances at the rich raw material nodes when RRi E Mr:
Gi =
E o~,jeo'
1,
Copper
r
e
c
o
v

in an etching plant
The purification task involves two rich streams (R1, R2) in copper. Mass flowrates and concentration specifications are given in Table 1 (E1Halwagi and Manousiouthakis, 1990). Two extractants are recommended for this separation task: LIX63(an aliphatic alphahydroxyoxime), L1, and P l ( a n aromatic betahydroxyoxime), L2. The data on copper concentrations and cost of the two available lean streams are given in Table 2.
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European Symposiumon ComputerAided Process Engineering6. Part A
Within the ranges of operating concentrations involved in this example, the copper transfer between the given rich and lean streams is governed by the following linear equilibrium relations: R1  L1 R1L2
: yl = 0.734xl + 0.001 : y1=0.111x2+0.008
R2  L1 R2L2
: y2 = 0.734xl + 0.001 : y2=0148x2+0.013
Two types of mass exchangers axe considered: a perforated plate column for LI(LIX 63) and a packed tower for L2(P1). There are two rich streams and two lean streams ( R i c h = {1,2}, L e a n = {1, 2}). The rich raw, rich product, lean raw, lean product and operating unit sets axe RR = {RRI,RR2}, RP = {RPI,RP2}, LR = LRI, LR2}, LP = {LPI, LP2} and O
=
011
=
{Oll,O12,O21,O22}, ({RR1,LR1},{RP1,LP1}),
0~2
=
({RRI,LR2}, {RP~,LP2}),
021
=
({RR2,LR1}, {RP~,LP1}),
O22
=
({RR2,LR2}, { R P 2 , L P 2 } ) .
Also M, R, and P are assigned by eqs. (2)(4). We input the M, R, P, O to the MSG (Friedler et al., 1993), the maximal structure of this example is shown in Fig. 4. In the MINLP based method (Papalexandri et al., 1994), 48 binary variables are used in this example, because they formulated the piping segments as well as mass exchangers as binary variables. In the worst case, 24s NLP should be evaluated in the MINLP based method. However only 9 feasible structures are evaluated from the maximal structure. Therefore the size of the search space is extremely reduced. The annualized investment cost of a perforated plate column is based on the number of plates N~, which is determined through the Kremser equation. The cost of a packed tower is based on the overall height H of the column. Annualized minimum composition difference of e = 0.0001 determines feasible mass exchange at the inlet and the outlet of each potential mass exchange unit. Investment cost functions (~1 and (I,~) are given in Table 3. The network structure and operating conditions of Pinch based method and those of MINLP based method are shown in Figures 5 and 6, respectively. Using the proposed method, we can get the better MEN structure (Fig. 7).
CONCLUSIONS A method using the Pgraph theory and the NLP formulation is proposed to find the optimal mass exchange network. The proposed two step procedure determines the network structure as well as
operating conditions. Pgraph theory minimizes the size of the optimization problem because it generates only feasible network structures. The significance of the proposed method is that it handles the combinatorial problem effectively and simplifies the optimization problems. ACKNOWLEDGEMENT Financial aid from KOSEF through Automation Research Center at POSTECH is acknowledged. REFERENCES E1Halwagi, M. et al., 1989, AIChE. J. 35, pp. 12331244. EIHalwagi,M. et al., 1990, Chem. Engng. Sci., 45, No. 9, pp. 28132831. Papalexandri, K. P., et al., 1994, Comput. Chem. Engng, 18, No. 11/12 pp. 11251139 Friedler, F. et al., 1992, Chem. Engng. Sci., 47, pp. 19731988 Friedler, F. et al., 1993, Comput. Chem. Engng., 17, pp. 929942. Friedler, F. et al., 1995, Chem. Engng. Sci., 50, No. 11, pp. 17551768. Table 1. Rich streams information of example 1 descrip. R1 R2
atom. solution rines water
Gi (kg/s) 0.25
y~
yu
0.13
0.1
0.10
0.06
0.02
Table 2. Lean streams information of example 1 descrip. L~ x~ x~ cost (kg/s) ($/kg) L1 LIX63 ~ 0.03 0.07 0.01 L2 P1 cx~ 0.001 0.02 0.12
Table 3. Investment cost data of example 1 Type perforated platecolumn(~l) packed column (~2)
Cost ($/yr) 600Ne°i74 7500(H) °'81
European Symposium on Computer Aided Process Enginearing6. Part A
$205
1o%
J
;0
,,,
.
o.,
03
,
6'~
61~33% !
Figure 1. Mass exchange network Pgraph of rich stream i and lean stream j.
io2.319%
.[
2.319%
2'X,
,,,~5,,
TAC=1.6317 x 15
oo~077 c,~,,~c~=,541
7,~ (
2~(~ ~Op.C~t= 1.51626x 10 5
Figure 5. Optimal MEN of example 1 by Pinch based method. .ME /
out
//'~z.7
(R ] ~ : ~
^bypass ~ bypass
/
/
"
"
\
p
\
•.
~/ \
, ,l
°27962
0,27_~2
10%
0.25
Figure 2. Variables of rich stream i.
3 2
~
bypass
''
/
t__
/
P ~
oo,,59
P
x ~ "1 ~'~'s Ix ~ •
~
J , ;7
O.Ol~59
Capital Cost = 9341
Figure 6. Optimal MEN of example 1 by MINLP based method.
Figure 3. Variables of lean stream j.
~~
RR I
RR2
3'~
0.1%
0.27~) 0,27962
LRI
LR2
' 
I
io~,
025 g)';
/
0.25
7.0'I
(}.01642 o.1%
o o96o:
T,I,
!SI
G
....
A
:
2.3 2',I, q).L
I~l~l
0.O6251 2.312% i
~ 2'~',
(
RP1
....
O
RP2
LP1
"
2'I, 0.01642 5 TAC=1.5338 x l0 CapitalCost= 3028
Op.Cos,=1.5035xIg
LP2
Figure 7. Optimal MEN of example 1 by proposed Figure 4. Maximal structure of example 1.
method.