SEPARATION-NETWORK SYNTHESIS:
GLOBAL OPTIMUM THROUGH
RIGOROUS SUPER-STRUCTURE
Z. Ercsey1, Z. Kovács1, F. Friedler1,2, L. T. Fan2
1
Department of Computer Science, University of Veszprém
Veszprém, Egyetem u. 10, 8200, Hungary
2
Department of Chemical Engineering, Kansas State University
Manhattan, Kansas, 66506, U.S.A.
Session on Process and Product Design
AIChE Annual Meeting
Miami Beach, Florida
November 15-20, 1998
OUTLINE
INTRODUCTION
CONVENTIONAL APPROACH
RIGOROUS SUPER-STRUCTURE
MATHEMATICAL PROGRAMMING MODEL
ALGORITHM FOR THE GENERATION AND
SOLUTION OF THE MODEL
EXAMPLES
CONCLUDING REMARKS
INTRODUCTION
 The algorithmic generation of the optimal
solution of a process synthesis problem
requires an
 appropriate
mathematical programming
model and a
 global optimization method.
 Various unexpected solutions obtained for
some simple classes of process synthesis
problems illustrate the difficulty in generating
a valid mathematical programming model:
 Kovács, Z., F. Friedler and L. T. Fan, Recycling in a
Separation Process Structure. AIChE J. 39(6), 1087-1089
(1993).
 Kovács, Z., F. Friedler and L. T. Fan, Parametric Study
of Separation Network Synthesis: Extreme Properties of
Optimal Structures. Computers chem. Engng 19, S465470 (1995).
 Kovács, Z., Z. Ercsey, F. Friedler and L. T. Fan,
Redundancy in a Separation-Network. Hung. J. Ind.
Chem. in press (1998).
CONVENTIONAL MATHEMATICAL
PROGRAMMING PROBLEM
INPUT
Mathematical programming
problem
(Objective function, constraints)
SOLUTION OF THE PROBLEM
Mathematical programming
method
?
Optimal solution
Comment: It is unsuitable for process synthesis.
PROCESS SYNTHESIS PROBLEM
(ALGORITHMIC APPROACH)
INPUT
Cost functions and constraints
for the operating units
and raw materials
Constraints for the product
MODEL GENERATION
(SYNTHESIS)
Generation of the mathematical
programming model
(MILP, MINLP, NLP)
?
?
SOLUTION (ANALYSIS)
Mathematical programming
method
Optimal solution
Comment: Model generation is the heart of a
synthesis problem
CONVENTIONAL "ALGORITHMIC"
METHODS FOR PROCESS SYNTHESIS
INPUT
Cost functions and constraints
for the operating units
and raw materials
Constraints for the product
MODEL GENERATION
Super-structure
generation
( MANUAL )
Super-structure
Model generation based
on the super-structure
( MANUAL )
MILP, NLP, MINLP
SOLUTION OF THE MODEL
Mathematical
programming
Process Network
Comment: The major activity is performed
manually.
PROBLEM DEFINITION

SNS problems with
 simple and sharp separators,
 dividers, and
 mixers.

Cost function
 The cost of a separation network is the sum
of the costs of its separators;
 The cost of a separator is: Difi
where fi mass load
Di degree of difficulty

Aim:
to generate the optimal structure.

Proposed method for SNS is
 Algorithmic in every step;
 Guarantees the optimality;
 Effective.
RIGOROUS SUPER-STRUCTURE

Let a set of operating units and the
mathematical model of each operating unit be
given.

A systematic procedure is presumed to be
available so that a valid mathematical
programming model can be generated for a
network of the given operating units.

A network of operating units is deemed to be a
rigorous super-structure if the optimality of the
resultant solution cannot be improved for any
instance of the class of problems by any other
procedure for network and model generation.

Note: a rigorous super-structure is not unique
since two different super-structures may lead
to an identical optimal solution for any
instance of a class of SNS problems.
ALGORITHM SNS-LMSG

Algorithm SNS-LMSG generates a rigorous
super-structure for SNS problems with simple
and sharp separators, dividers, and mixers, in a
finite number of steps where the cost of a
network is the sum of the separators' costs,
each of which is proportional to its mass load.

It is illustrated with examples.
ILLUSTRATIVE EXAMPLE FOR
GENERATING
RIGOROUS SUPER-STRUCTURE
Feed-stream 1
Feed-stream 2
Product-stream 1
Product-stream 2
Product-stream 3
A
A
A
-
Components
B
B
B
B
-
C
C
C
C
Step 1: Creating and linking a divider to each
feed-stream and a mixer to each productstream.
[A,B,0]
M1
[A,B,C]
D1
[0,B,C]
M2
[A,B,C]
D2
[0,0,C]
M3
Step 2: Creating and linking separators S11 and S12
to the outlet of divider D1.
1
S1
[A,B,0]
[A,B,C]
M1
D1
S21
[0,B,C]
M2
[A,B,C]
[0,0,C]
D2
M3
Step 3: Creating and linking separators S21 and S 22
to the outlet of divider D2.
[A,B,0]
S11
M1
[A,B,C]
D1
S21
[0,B,C]
M2
S12
[A,B,C]
D2
[0,0,C]
S22
M3
Step 4. Creating and linking dividers D3 and D4
to the outlets of separator S11.
D3
[A,B,0]
S11
M1
D4
[A,B,C]
D1
S12
[0,B,C]
M2
S21
[A,B,C]
D2
[0,0,C]
S22
M3
Step 5. Establishing a bypass from the outlet of
divider D3 to the inlet of mixer M1 for
product-stream [A,B,0].
[A,B,0]
D3
M1
S11
D4
[A,B,C]
D1
S12
[0,B,C]
M2
S21
[A,B,C]
D2
S22
[0,0,C]
M3
Step 6. Creating and linking separator S 32 to the
outlet of divider D4.
[A,B,0]
D3
M1
S11
D4
[A,B,C]
S32
D1
S12
[0,B,C]
M2
S21
[A,B,C]
D2
S22
[0,0,C]
M3
Step 7. Establishing a bypass from the outlet of
divider D4 to the inlet of mixer M2 for
product-stream [0,B,C].
[A,B,0]
D3
M1
S11
D4
[A,B,C]
S32
D1
S 21
[0,B,C]
M2
S12
[A,B,C]
D2
S22
[0,0,C]
M3
Step 8. Creating and lniking dividers D5 and D6
to the outlets of separator S 32 .
[A,B,0]
D3
M1
D5
S11
D4
[A,B,C]
S32
D1
D6
S12
[0,B,C]
M2
S12
[A,B,C]
D2
S22
[0,0,C]
M3
Step 9. Establishing two bypasses from the outlet
of divider D5, one to mixer M1 for
product-stream [A,B,0] and the other to
mixer M2 for product-stream [0,B,C].
[A,B,0]
D3
M1
D5
S11
D4
[A,B,C]
S32
D1
D6
S21
[0,B,C]
M2
S12
[A,B,C]
D2
S22
[0,0,C]
M3
Resultant rigorous super-structure.
[A,B,0]
M
D
D
1
S
S2
D
D
[A,B,C]
D
D
D
S1
S2
D
D
D
[0,B,C]
M
S1
D
D
S2
D
[A,B,C]
D
D
D
S2
S1
D
[0,0,C]
D
M
GENERATION OF THE MATHEMATICAL
PROGRAMMING MODEL
 The mathematical programming model
derived from the rigorous super-structure
should be as simple as possible without
impairing the optimality of the resultant
solution.
 Literature
 Bilinear (nonlinear programming)
Quesada and Grossmann (1995)
 General nonlinear programming
Floudas (1987)
 Benders decomposition
Floudas and Aggarwal (1990)
 Present work
 Linear programming
 Generates the global optimum
MATHEMATICAL PROGRAMMING MODEL
 Based on the rigorous super-srtucture a linear programming
model can be generated

min  (d i
iS
n
 ( x ji    kic f kc ))
 j :( j ,i ) A c 1k F
subject to
 xij
1
(i, j )  A where i  D
i  D where l  F such that l , i   A
 xij
 x kl
(k , l )  A where i  D such that (l , i )  A
0  xij
{ j :( i , j )A}
{ j :( i , j ) A}
pic 
( xlj  kic f kc )
{( l , j ):( j , i ) A}
k F


i  P and c  1,2....,n
1 there is a path from node k to node i with component c

otherwise
0
k  F, i  P  S
 kic
ILLUSTRATION OF THE MODEL
 Splitting ratio
xD2M3
x D1M1
[a1,a2,a3,a4,a5,a6]
D1
xD1S5
x D1S3
x
x
x
x D2S2
S3
x D1S3
1  x D1M1  x
x
D2
x D1S3
x D2S2
D1S 3
x
xD3M1
x D3S4
xD3M2
D1S 5
D1S 3
 x D2 M 3  x
D1S 3
 x D3M1  x
D2 S 2
 x D4 M1  x
D2 S 2
 x D5 M 2  x D5 M 4
D2 S 2
D3S 4
 x D3M 2
D4 S 1
x D4S1
S2
x D2S2
D3
D4
xD M
4 1
D5
x D5M2
x D5M4
ILLUSTRATIVE EXAMPLE
 Problem Specification (Quesada and
Grossmann, 1995)
Component
Feed-stream
Product-stream 1
Product-stream 2
A
10
6
4
B
10
4
6
C
10
2
8
Degree of difficulty of each separator is 1.
 Rigorous super-structure generated by
Algorithm SNS-LMSG
x D1M1
x D2M1
D2
x D2M2
x D6M1
S11
D6
x D3M1
x D6M2
x D3S22
x
[6,4,2]
S22
D3
D1S11
M1
x D7M1
x D8M1
x D3M2
D7
x D7M2
[10,10,10]
D1
D8
x D4M1
x
x D8M2
D1S21
S12
D4
x D9M1
x D4S12
S21
D 9 xD9M2
x D4M2
x
D5
D5M1
x D5M2
x D1M2
[4,6,8]
M2
 Mathematical programming model: LP
min 30  xD S 1  30  xD S 2  20  xD
1 1
1 1
2
3S2
 20  xD
1
4 S2
subject to
i, j  {D1M 1 , D1M 2 , D1S11 , D1S12 , D2 M 1 , D2 M 2 ,
0  xij
D3 M 1 , D3 S 22 , D3 M 2 , D4 M 1 , D4 S 21 , D4 M 2 ,
D5 M 1 , D5 M 2 , D6 M 1 , D6 M 2 , D7 M 1 , D7 M 2 ,
D8 M 1 , D8 M 2 , D9 M 1 , D9 M 2 }
xD1M1  xD S1  xD S 2  xD1M 2  1
1 1
1 1
x D2 M1  x D2 M 2  x D S1


x D3M1  x D S 2  x D3M 2  x D S1 
3 2
1 1

x D4 M1  x D S1  x D4 M 2  x D S 2 
4 2
1 1

x D5M1  x D5M 2  x D S 2

1 1

x D6 M1  x D6 M 2  x D S 2

3 2

x D7 M1  x D7 M 2  x D S 2

3 2

x D8 M1  x D8 M 2  x D S1

4 2

x D9 M1  x D9 M 2  x D S1

4 2
1 1

4  10  x D M
2  10  x D M
4  10  x D M
6  10  x D M
8  10  x D M
6  10  x D1M1  x D2 M1  x D4 M1  x D8 M1


1
1
 x D3M1  x D4 M1  x D6 M1  x D9 M1
1
1
 x D3M1  x D5M1  x D7 M1
1
2
 x D2 M 2  x D4 M 2  x D8 M 2
1
2
 x D3M 2  x D4 M 2  x D6 M 2  x D9 M 2
1
2
 x D3M 2  x D5M 2  x D7 M 2














 Optimal structure
x D1M1
D2
x D2M1
[6,4,2]
S11
x D1S11
[10,10,10]
M1
D3
x D3M2
D1
x D1S21
D4
x D4M1
S21
M2
D5
xD5M2
xD1M2
x D1M1  0.2
x
D1S11
 0.2
 0.2
x D1M 2  0.4
x D2 M1  0.2
x D4 M1  0.2
x D3M 2  0.2
x D5M 2  0.2
x
D1S12
The value of the cost function is 12.
[4,6,8]
ADDITIONAL EXAMPLES
 Example 1 (Quesada and Grossmann, 1995)
 4 components,
 3 feed-streams,
 3 product-streams
 Example 2 (Quesada and Grossmann, 1995)
 6 components,
 1 feed-stream,
 4 product-streams
EXAMPLE 1
 Problem specification
Component
Feed-stream 1
Feed-stream 2
Feed-stream 3
Degree of Difficulty
Product
Product-stream 1
Product-stream 2
Product-stream 3
A
6
8
0
B
4
6
0
4
C
0
10
5
1.5
D
0
6
5
4
Sum of the
components
Component
information
15
20
15
A9 B3 C3 D=0
B7 C7 B=C
D9 A=0
 Best known solution (Quesada and Grossmann, 1995)
[10,3,2,0]
M
4.5
[6,4,0,0]
1.788
D
3.722
13
8.479
M
S1
0.944 B
D 2.544
[8,6,10,6]
D
[4,7,7,2]
S2
M
4.8
D
17
[0,0,5,5]
8.8
S3
3.4 D
1.6 CD
[0,0,6,9]
M
Comment: The cost of the objective function is 138.18.
 Optimal solution based on rigorous super-structure
0.75
[10,3,2,0]
M1
[6,4,0,0]
D1
0.25
[8,6,10,6]
S11
S12
0.333
[4,7,7,2]
D2
M2
0.5
D3
0.2
0.167
D5 0.367
0.667
S2
0.567
S3
D4
[0,0,5,5]
0.1
[0,0,6,9]
M3
Comment: The cost of the objective function is 104.26.
 Optimal solution with combined separators (based on rigorous
super-structure)
[4.5,3,0,0]
[10,3,2,0]
M2
[6,4,0,0]
D1
[1.5,1,0,0]
M1
[0,0,2,0]
S1
[4,3,0,0]
[8,6,10,6]
[2.667,2,3.333,2]
[4,7,7,2]
D2
M3
D3
[1.333,1,0,0]
D5
[0,0,3.667,0]
[5.333,4,6.667,4]
S2
[0,0,5.667,3.4]
S3
D4
[0,0,5,5]
[0,0,1,0.6]
[0,0,6,9]
M4
Comment: The cost of the objective function is 104.26.
EXAMPLE 2
 Problem specification
Component
Feed-stream
Product-stream 1
Product-stream 2
Product-stream 3
Product-stream 4
Degree of Difficulty
A
23
3
8
5
7
B
19
2
10
4
3
1.5
C
25
6
8
10
1
3.0
D
21
8
8
3
2
2.0
E
26
4
6
11
5
2.5
F
26
10
5
4
7
4.0
 Solution
 Best known solution: 388.00.
 Optimal solution based on the rigorous
super-structure: 330.76.
 Optimal solution
0.105
[3,2,6,8,4,10]
M1
0.028
D2
0.025
0.049
0.007
D7
0.021
0.146
S2 1
S2 4
0.178
S1
[8,10,8,8,6,5]
M2
D3
0.011
S2 2
0.141
0.068
[23,19,25,21,26,26]
D1
0.257
S3
D8
0.022
0.163
D4
0.038
S4 2
S5 3
S5 1
0.192
0.143
0.128
0.086
S2 3
[5,4,10,3,11,4]
D5
0.214
M3
0.011
S4 1
D6
0.106
S5 2
0.029
0.097
D9
0.055
0.04
0.077
[7,3,1,2,5,7]
M4
SUMMARY OF THE EXAMPLES
 Example 1
Method
Quesada and
Grossmann (1995)
Present work
Number of Type of Optimal
variables
model
solution
113
nonlinear 138.7
90
linear
104.2
Computational
time (sec)
0.77*
0.34**
 Example 2
Method
Quesada and
Grossmann (1995)
Present work
 IBM RS600/530
 100MHz Pentium PC
Number of Type of Optimal
variables
model
solution
430
nonlinear 388.0
1094
linear
330.7
Computational
time (sec)
33.0*
1.6**
CONCLUDING REMARKS
 A new method has been presented for
separation network synthesis:
 It is algorithmic in each step.
 It is based on the rigorous super-structure.
 A linear programming model is generated.
 The optimality of the solution is guaranteed.
 Several examples illustrate the efficacy of this
new method.