zhou 1988

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Comput. them. Engng, Vol. 12, No. 6, pp. 581-588, 1988

Printed in Great Britain. All rights rcscrvcd

@098-1354/88 3.00 +O.OO

Copyright 0 1988 Pergamon Press plc

A NEW STRATEGY OF NET DECOMPOSITION

IN PROCESS SIMULATION

ZHOU

Lr,t

HAN ZHJMWEI

nd Yu KUOTXJNG

Chemical Engineering Research Center, Tianjin University, Tianjin, China

(Received 12 June 1986; final revision received 13 November 1982;

received

or

publication 1 December 1987)

Abstract-A new strategy of finding a set of tear streams is proposed in this paper. It includes lining-up

all the nodes of a given net in the declining order of node weights, and then adjusting node positions in

the sequence, first from left to right, and then from right to left to minimize the total weights of the

remainingcounter streams. The resultingnode sequence indicates the computation order of the nodes,

and still the existing counter streams constitute the tear set. Although the optimality of the strategy seems

difficult to be mathematically proven, it produced optimal solutions for all the well-known examples

tested. Another prominent feature of the method lies in the simplicity of its logic in that neither loop

identification nor excessive matrix operations are needed. A real process flowsheet even as complex as a

heavy-water plant, can be rapidly decomposed on a microcomputer.

INTRODUCTION

Net decomposition is a classical problem in process

simulation. A sequential process simulator trans-

forms a process configuration into an information

flow diagram, which is called a network, or a net if

it contains loop(s). Tearing loops is the first step in

the simulation of a complicated process, and it is an

important factor in determining the speed of con-

vergence, and consequently in costing the simulation.

Since the early days of process simulation, therefore,

many works have been dedicated to this topic.

Net decomposition, however, is one of the most

difficult operations in this field. Although many

strategies have been developed and applied in

commercial simulators, the effect of tearing or the

convergence property is still not fully understood

(Motard et

al.,

1975; Chen and Chen, 1979). As a

consequence, it is difficult to evaluate a strategy

precisely.

The following debatable criteria have been widely

accepted by many authors as a basis to develop

decomposition methods. It is argued that:

1. The fewer the tearing operations on a loop, the

more rapidly the iteration procedure converges (Chen

and Chen, 1979).

2. The smaller the total weights of the tear

streams, the lighter the computation load.

The weight of a stream generally equals the num-

ber of variables carried in the stream, but it can also

be a relative coefficient assigned to the stream accord-

ing to the ease of convergence in the preliminary

computations.

The aim of net decomposition is to recognize the

best set of tear streams, and to determine the per-

tTo whom all correspondence should be addressed.

mutation sequence of the nodes. The preferability of

a tear set could be tentatively judged by utilizing the

above criteria. However, a given tear set can have

several different node sequences. The sequence order-

ing is of no consequence in simutaneous solution

strategy, but it may affect the convergence property

of a nested strategy. This effect, however, could not

be determined without elaborate trial computations.

Therefore, all the possible node sequences of a given

tear set were considered virtually equal.

For most of the published decomposition methods

(Barkley and Motard, 1972; Forder and Hutchinson,

1969; Gundersen and Herzberg, 1982; Himmelblau,

1966; Pho and Lapidus, 1973; Yu, 1980), the aim was

achieved in two phases, i.e. first, identification of the

loops of the given net and then selection by some

method of a set of streams to tear. The whole

decomposition process involved excessive manipu-

lations of the information flow diagram or its dual

fotm of signal flow and tedious operations on the

adjacency matrix. Furthermore, some methods were

inefficient when applied to particular cases.

It will be shown in this paper that the new strategy

succeeded in attaining the minimum sum of the

weights of the teared streams, whilst ensuring that the

number of loops tearing was as low as possible. In

addition, this method yields a tear set and a feasible

node sequence simultaneously, without the need for

loop identification, matrix operations, or excessive

graph simplifications. The simple logic of the method

allows a complicated net to be decomposed manually

or more efficiently on a microcomputer.

THE NEW DECOMPOSITION METHOD

The essentials of the new decomposition method lie

in the transformation of the 2-D information flow

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Fig. 1. AUN definition.

graph

into

a 1-D node sequence, followed by individ-

ual adjustment of the node positions in the sequence.

The detailed procedure

is

as follows.

1. Simplification of the given graph. The nodes

without input-output streams, or those having one

input and one output stream of equal weight may

be disregarded from the consideration of decom-

position. This is because they neither affect the tear

set nor the final node sequence Each node in the

simplified graph is connected with at least two other

nodes.

2. Formulation of the node adjacency matrix of

the simplified graph. The element corresponding to

the ith row

and

jth column equals the weight of the

i -j stream. Defining

Fw,=w0,

the output weight of node

i;

cwij= +(j),

the input weight of node j; then the node weight is

defined as w(i) = w (i) - w + (i).

3. Formation of an initial node sequence (INS)

from the adjacency matrix in the order of decreasing

node weight. A duplicated INS will never change, and

is known as a reference node sequence (RNS).

4. Individual inspection and adjustment of node

positions in the INS In order to explain the adjust-

ment of the node positions, some conceptions are

defined below.

(a) Dir ect streams (DS) and counter streams (CS)

Since the node sequence defines the direction of

information flow, and a stream with the same direc-

tion as the sequence causes no information feedback,

this is defined as a DS. On the other hand, a stream

pointing to an upstream node represents a feedback

flow and is thus defined as a CS, which is to be

changed to direct if possible.

(b) Active upstream node (AUN)

For a given node L and a given CS L + G, if there

is a node I between L and G (Fig. 1) that emits a DS

to node

L,

the intermediate node I is defined as an

active upstream node (ALIN) of L.

(c) Counter weights (CW) and dir ect weights (DW)

For a given node L, the sum of the weights of the

DSs terminating at L is called the DWs (DW) of node

L ;

the sum of the weights of the CSs emitted fro ZDW, then the inspected

node (also, the origin of the CS) is moved to the

immediate left of the target node of the CS with a

consequent decrease in the net CW (if the node has

more than one CS, then it is moved to the immediate

left place of the target node of the CS that reaches far

most left); if XW d XDW, then the origin node of

the CS is first moved to the immediate right of the

AUN that locates nearest to the inspected node. The

two nodes from a node block (NB), which is sub-

sequently treated as a new node. The node block can

either move left to change the remaining CSs direct,

or grows up to include the next nearest AUN. If the

target node of a given CS is contained in the NB, then

the CS cannot be converted, and all the CSs con-

tained in the NB are stored in an array of the tear set.

When the last node of the RNS has been inspected,

a preoptimal node sequence and tear set are obtained.

5. Reversal of the preoptimal node sequence and

reexamination of it in similar manner, but with DSs

changing directions. The backward inspection starts

from the last node of the inversed sequence and goes

left until the second node of it. The necessity of

implementing this step stems from the fact that the

first node in RNS has not been inspected hitherto,

and its position might need to be adjusted. Fig. 2

provides an example of such cases. As shown in Fig.

2a, no one node can be moved before node D;

however, if we reverse the sequence and begin to

change the DSs over, then we should move node A

to the left of node D because the DWs of C + A and

(b)

Fig. 2. Consequence of node sequence inversion.

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A new strategy in process simulation

Form initial node sequence

INS and RNS. Begin position

insoection at 2nd node in RN.5

I

1

any

CS

of the

Move the

inspected node to

the ILP of its most left

TN of the Css to change

them direct

Move the inspected node to

the IRP of its most right

AUN to form a NEI that to

be treated as a new node

Reverse the preoptimal

node sequence and

adjust node positions

once more

inverse node sequence

again and

insert the

nodes omitted before

583

AUN = Active upstream node

cs

= Counter stream

cw

= Counter weights

DW

= Direct weights

ILP

= Immediate left place

IRP = Immediate right place

NB = Node block

TN = Target node

Fig. 3. Flow-chart of new method.

D + A are greater than the CW of A + B

Fig.

2b)

(note: the CSs

A + E

and

A -P F

are not

active to

stream D + A .

As a result, the number of tear

streams is decreased by 1 in comparison to the

preoptimal sequence.

6. Reversal of the node sequence again and in-

,wrtion of the omitted nodes following their upstream

nodes. The final node sequence and tear set are thus

obtained.

Fig. 3 illustrates the flowchart of the computer

program, wtitten for the aforementioned operations.

The new strategy was tested on nine typical problems

and the results of these decompositions are listed

below.

CASE STUDIES

Nine well-known examples were collected to test

the new strategy. To illustrate the details of the

method, two examples hake been manually decom-

posed; the remaining exampleswere decomposed on

a microcomputer.

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w j)=w- j)-w+ j).

v, v,

v, v, vs

w -

K

V*

v3

V4

VS

W+

W

r

2

2 T

3

1

4

7 1 8

1

3

2 6

4 5

9

8 8 3 7 3

_ -6 -4

5

-1 6

Arranging nodes in the decreasing order of node

weights, the initial node sequence (as well as the

reference node sequence) is obtained as follows:

V,,

v3, v,, v*, VI.

1. Rubin graph (Rubin , 1962)

Figure 4 cannot be further simplified; therefore, we

directly form its node adjacency matrix. The

i ,

j)th

element in the matrix that follows is the output

weight of node i with respect to node j. The last

element of the ith row is the total output weights of

node i; while the element before last in thejth column

is the total input weight of node j. The last element

in the column is the node weight of j, i.e.

Now, the position of node V3 s considered and any

necessary adjustments to this are determined. All the

output streams of V are enumerated and the counter

ones noted.

V, +

V,

is a counter stream. It can be seen from the

V,

h column of the matrix that

Vs

s not an AUN of

V,; therefore, V3 s moved to the left of V, and the

node sequence becomes:

v,, vs, v,, Vz, V,.

The next node of RNS to be inspected is V,.

V,+ V,

is a CS.

V,

is not an AUN of

V,

either, so

the stream can also be changed to a direct one. The

node sequence is then:

V,, V,, Vs. V2, V,.

The next CS is V,+ V,. It is learnt from the V,th

column of the matrix that V, and Vs are AUNs of V,

and the total DWs (ZDW = 8) are greater than the

CWs (ZCW = 1) of node V,; therefore, a NB

(V, V, V,) is formed. The CW of the NB (1) is again

less than ZDW(7 + 1 = 8) of it;

(1)

1

V,- (V,V,V*)V,

(1)

=9

21

1)

Fig.

4. The Rubin graph (the number

in parenthesis beneath

a

stream line is the weight of the stream).

ti

Fig. 5. The reverse of the preoptimal node sequence.

hence. the NB grows to include node

V3;

therefore,

the CS V, + V, cannot be changed to direct. Simi-

larly, the fIna1 CS

V, + V,

cannot be changed to

direct. The preoptimal node sequence has thus been

produced. The preoptimal node sequence is now

reversed (Fig. 5) and reexamined for possible con-

versions of DSs. However, no more adjustments

could be made. The final node sequence shown below

was obtained after once more reversing the node

sequence:

SIlll

1

I

v,...+v,..-+v,..

.-+v*..

. v,

t

SZf2)

I

The tear

set = {S,, S,).

The total weights of the set = 3.

A feasible computation order is shown by dashed

arrows.

2.

Forder and Hutchison graph (Forder and Hutchi-

son, 1969)

Apparently, no one node can be skipped. The

following is the node adjacency matrix.

A

B

c

D

E

F

Wf

W

A

B C D E F w-

4

4

5

8 13

4

10 14

6

2 4 12

3

2 5

2 2

15 7

10 10 4 4

.-I1 6

4 2 1 -2

According to the decreasing ordei of ws, we form the

initial node sequence as:

B, C, D, E, F , A.

The procedure of inspecting node positions begins

at the second node, i.e. C. It is learnt from the Cth

row of the matrix that node C does not have any CSs;

Fig. 6. Forder-Hutchison graph.

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A new strategy

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585

neither does node D. However, node E has two CSs,

E +C

and

E -+B. So, we

check if there are any

AUNs of node

E

that locate between

(2)

1

B C D&E

t

(3)

E and B and emit streams to E.

It

is seen from the

Eth column of the matrix that node D emits a stream

to

E

with weight 2. Because the sum of CW (2 + 3)

is greater than DW (2), node E is moved to the

immediate left of node B with the consequence of

decreasing the net CW by 3. The node sequence now

becomes E, B, C, D, F, A.

Node

F

has a CS

(F-P E)

with weight 2. However,

it cannot be changed since there is an AUN

D

with

DW of 4.

Finally, node

A

has a CS

(A + B)

with weight 4.

This CS cannot be changed either because the sum of

DW of DSs (B + A, C + A, D + A) is greater than

CW (4). The preoptimal solution is thus obtained,

and is shown in Fig. 7a. Figure 7b is its inversed form.

There are two DSs of node E, D + E and F + E,

with the total weights of 4. The CSs that emit from

E

and terminate on nodes between

E

and

Fare E + B

and E +C with total weights of 5.

Since

ZCW > ZDW, the DSs cannot be changed.

The next node to be inspected is

B.

It has a DS

A + B with weight 4, and two CSs B-P A and B + C

with total weights of 13; therefore,

Bs

place cannot

be changed either. Nodes C,

D, F

do not have any

DSs coming up from left and hence cannot be further

moved. The inspection procedure has thus been

finished.

Reversing the node sequence again, we obtain the

final solution:

Node sequence =

E, B, C, D, F, A.

(a)

Fig. 7a. The preoptimal sequence of Example 2.

Fig. 7b. The inversed sequence of Fig. 7a the backward

inspection starts from E).

Fig. 8. Christensen-Rudd graph.

The tear set = {S,, S,, S,}.

The total weights of the set = 8

In comparison, the solution achieved by Forder

and Hutchison (1969) was:

Node sequence = B, C, D, F, E, A.

The tear set = {S,, S,, S,}.


3.

Chri stensen-Rudd graph (Chr istensen and Rudd,

1969)

The initial node sequence is:

A, E , C, B, D.

The final result as indicated below can be easily

obtained by the aforementioned procedure:

.

ATE B

AD

S6

The set of tear streams =

{S, , S,}


4. Upadhue-Grens graph (Upadhue and Grens, 1972)

The final node sequence =

D, C, A, B.

The tear set = {S,, S,}.


5. Chr istensen-Rudd graph No. I (Chr istensen and

R&d, 1969)

Figure 10 is a nonweighted,graph (i.e. every stream

is of weight one) and it can be simplified as shown in

Fig. 11.The node weights were determined from the

node matrix of the simplified graph (not shown). The

initial node sequence is then obtained by arranging

the nodes in the declining order of node weights.

Adjustment of the node positions was carried out

according to the aforementioned rules. Thus the final

node sequence was obtained:

1, J, Q, P, U, X, y, N, M, 0, E, S, T, A, E, C, i,

E, F, L, K, G, H, V, W.

The tearset = {S,,, S,,, S,>-


q

3)

Fig. 9. Upadhue-Grens graph.

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Fig. 10. Christensen-Rudd graph No. 1.

Fig. 11. Simplified Christensen-Rudd graph No. 1

6. Chri stensen-Rudd graph No. 2 (Chr istensen and

Rudd, 1969)

The

original and simplified graphs of C and

R

No. 2 are shown in Figs 12 and 13, respectively. The

final node sequence is;

I,

J, K, A, T, BB, AA , CC, M , B, C, D, N, 0, P,

Q, R, W, x, Y, Z DD, V, U, L, E, S, F, G, H.

The tear set = {S1,,

SIP,S,}.

This is the minimum number of tears, and one

less than the method proposed by Gundersen and

Herzberg (1982).

Fig. 12. Christensen-Rudd graph No. 2.

Fig. 13. Simplified Christensen-Rudd graph No. 2.

7. Sargent and Westerberg graph (Sargent and

Westerberg, 1964)

Figures 14 and 15 are the original and simplified

graphs, respectively. There are two parallel streams

between

D

and

E, F

and G in Fig. 15. They are

merged to one stream because they neither form a

loop nor change the direction of information flow.

the final sequence is:

I, D, K, F, A, B, E, L, Q, S, R,.M , C, G, H , N, 0,

P, J.

The tar set = {S,, S8, S,,, , Sz8, J:


8. Shannon graph (sulphur ic acidplant; Shannon et al.,

1966)

There are 41 nodes, 61 edges and 103 cycles in the

net. It is a fairly complicated graph, yet it can still be

decomposed manually. The final node sequence out-

put from a computer was:

25, 26, 27, 28, 3, 4, 5, 9, 38, 41, 42, 43, 44, 36, 9,

39, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,

32, 33, 34, 35, 22, 23, 31, 37, 4 1, 24.

Fig. 14. Sargent-Westerberg graph.

Fig. 15. Simplified Sargent-Westerberg graph.

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Fig. 16. The information flow graph of a sulphuricacid plant.

The tear set={s,,s,,,s,,,s,,,s,}.


9. Gun&men and Herzberg graph (heavy water plant;

Gundersen and Herzberg, 1982)

This is a very complex graph comprising 109

nodes, 163 edges and 13,746 cycles, and was sug-

gested as a benchmark for decomposition algorithms

by Gundersen and Herzberg (1982). It was also

decomposed by hand using the proposed method,

but the following are the results of the significantly

more rapid decomposition, performed on a micro-

computer.

The final node sequence is:

64, 105, 93, 92, 91, 90, 99, 87, 71, 70, 84, 83, 82,

81, 85, 79, 104, 103, 100, 86, 106, 101, 96, 95, 102, 80,

97, 62, 68, 66, 42, 41, 40, 39, 57, 55, 54, 53, 52, 51,

50, 77, 98, 107, 109, 108, 88, 89, 78, 69, 6, 5, 36, 37,

35, 15, 14, 13, 12, 29, 27, 26, 33, 32, 31, 30, 56, 48,

45, 46, 4, 3, 2, 1, 28, 20, 18, 19, 7, 47, 65, 76, 75, 74,

49, 67, 63, 61, 60, 59, 58, 73, 72, 94, 25, 24, 23, 22,

34, 21, 44, 38, 17, 16, 11, 8, 9, 10, 43.

Fig. 17. Simplified Shannon graph.

-he

tear=t =

{SW

SW. Sl14, S,,, Sll, Sls, SIM.

s,.

s71,

s,,,, *r ,,J.

The total weights of the set = 12:

This is the minimum number of tears. The method

proposed by Gundersen and Herzberg (1982) gave 13

tears.

CONCLUSIONS

The new decomposition method is creditable firstly

for its simple logic and operations. Manual decom-

position even for complicated graphs is possible

albeit a lengthy process; however, decompositions on

computer are more efficient and reliable. A math-

ematical proof that the strategy always guarantees

optimal solutions is at present not performed, but this

method gave optimal solutions for all the cases

studied.

No comparison in terms of computer time with

the other decomposition methods was attempted;

however, this new method is substantially easier to

program than those methods based on graph

manipulations, and works much faster than those

based on matrix operations. Furthermore, most pre-

vious methods failed to produce optimal solutions for

all the problems cited.

REFERENCES

Barkley R. W. and R. L. Motard, Decomposition of nets,

Chem. Eng ng J. 3, 265 1972).

Chen M. and L. Chen, Decomposition of nets--the criteria

of optimal tearing. J.

Shanghai Inst. Chem. Technol. 1-2,

11 1979).

Christensen J. H. and D. F. Rudd, Structuring design

computations. AfChE J f 15, 94 1969).

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Forder G. J. and H. P. Hutchison, The analysis of chemical

plant flowsheets. Chem. Enana Sci. 24. 771 (1969).

Gundersen T. and T. Her&&g, partitioning and tearing

chemical process flowsheets. Proc. Int. Symp. Process

Syst. Engng, PSE 82, Kyoto, Japan, Vol. 2, p. 9 (1982).

Himmelblau D. M., Decomposition of large-scale systems-

1. Svstems comaosed of lumocd uarameter elements.

Ch . Engng SC;. 21, 425 (1986). A

Motard R. L., M. Schacham and E. M. Rosen, Steady-state

Chemical process simulation. AZChE JI 21, 417 (1975).

Pho T. K. and L. Laoidus.

ToDi S

in cornouter-aided desian:

part I: an optimum tearing algorithm for recycle systems.

AIChE JI 19, 1170 (1973).

Rubin D. I., Generalized material balance. Chfm. Engng

Prog. Symp. Ser. 58, 43 (1962).

Sargent R. W. and A. W. Westerberg, SPEED-UP in

chemical engineering design. Trans Inst. Gem. Engrs 42,

T190 (1964).

Shannon P. T. er al., Computer simulation of a sulphuric

acid plant. Chem. Engng Prog. 62, 49 (1966).

Upadhue R. S. and E. A. Grens, An efficient algorithm for

optimum decomposition of recycle systems. AIChE JI 18

465 (1972).

Yu Zhongming, The nodes sequence minimized total recycle

weight in weighted directed graph. J. Shanghai Inst.

Chem. Technol. 1, 111 (1980).

zhou 1988

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