1. SOME RESULTS IN GRAPH THEORY
1 ) A graph is any connection of points, some pairs of which are
connected by lines.
2 ) If a graph has p points and q lines, it is called a (p,q) graph.
points process and utility streams
lines heat exchangers
3 ) A path is a sequence of distinct lines, each are starting where
the previous are ends, e.g. AECGD in Fig. A. A B C D
Figure A
Figure B E F G H A B C D E F G H

2. SOME RESULTS IN GRAPH THEORY
4 ) A graph is connected if any two points can be joined by a path,
e. g. Fig. A
5 ) Points which are connected to some fired point by paths are said
to form a component, e. g.
Fig A has one component.
Fig B has two components.
6 ) A cycle is a path which begins and ends at the same point, e. g.
CGDHC in Fig. A.
7 ) The maximum number of independent cycles is called the cycle
rank of the graph.
8 ) The cycle rank of a (p,q) graph with k components is
q - p + k

3. An Implication of Cycle Rank
U = N+L-S
where,
N = the total number of process and utility streams
L = the number of independent loops
S = the number of separate components in a network
U = the number of heat exchanger services

4. Examples of U = N+L-S U = N-1 = 5 U = N-2 = 4 U = N+1-1 = N = 6 ST H1 H2
U = N-1
= 5
U = N-2
= 4
U = N+1-1
= N
= 6 C1 C2 CW ST H1 H2 C1 C2 CW ST H1 H2
X X
30-X X C1 C2 CW

5. CAPITAL TARGET Umin = N - 1 where,
Umin = the minimum number of services
N = the total number of process and
utility streams
Note,
U = N + L – S
L = 0
S = 1

6. § PINCH DESIGN METHOD
RULE 1: THE “TICK-OFF” HEURISTIC
UMIN = N-1
- THE EQUATION IS SATISFIED IF EVERY MATCH
BRINGS ONE STREAM TO ITS TARGET TEMPERATURE
OR EXHAUSTS A UTILITY.
- FEASIBILITY CONSTRAINTS :
ENERGY BALANCE
TMIN

7. Example 1
Stream No TS TF CP Heat Load
and Type (F) (F) BTU/hr F Q BTU/hr
(1) Cold
(2) Cold
(3) Hot
(4) Cold
(5) Hot
(6) Cold
(7) Hot
Tmin = 20F
Qhmin =  104 BTU/hr
Qcmin = 0

8. Hot streams
CP Q
1.32
2.624
590 
471 419 
533 
400 
430 
400 
280  3 ○ 5 ○
505.6 7 1
416 ○ 2
505.6 4 ○○
341.1 6
341.1
Cold streams

9. CP Q
1.557
4.128
590 574 
471 
400 
430 
400  ○ 3 ○
86.3 5
254 ○ 1
86.3 2
412.8 ○ 4
412.8

10. CP Q
590  
400  
430  ○ ○ 3 ○ 1 H ○ 2
22.4

11. CP Q
1.32
2.624
590 
471 
533 
400 
430 
400 
280  ○ ○ ○ 3 ○ ○ 5 ○
505.6 7 ○ ○ H 1
86.3 ○ ○ 2
22.4
505.6 ○ 4
412.8
341.1 ○ 6
341.1

12. § PINCH DESIGN METHOD
RULE 2: DECOMPOSITION
THE HEN PROBLEM IS DIVIDED AT THE PINCH INTO SEPARATE DESIGN TASKS.
THE DESIGN IS STARTED AT THE PINCH AND DEVELOPED MOVING AWAY FROM THE PINCH.

13. EXAMPLE 2 Temperature Heat Capacity Supply Target Flowrates Heat load
Process Stream TS TT CP Q
no. Type (F) (F) (104 BTU/h/F) (104 BTU/h)
1 Cold
2 Hot
3 Cold
4 Hot
Tmin = 10 F
QHmin = 50  104 BTU/h
QCmin = 60  104 BTU/h

14. PINCH DECOMPOSITION DEFINES THE SEPARATE DESIGN TASKS!
260    2
250    4
240    1
240  3 C
= 60 Btu/h H
= 50 Btu/h
Umin = 4
Umin = 3
PINCH DECOMPOSITION DEFINES THE SEPARATE DESIGN TASKS!

15. BELOW THE PINCH
CP Q
190  2 3
190   4 4 G 60
190   3 4 1
ABOVE THE PINCH
CP Q
260  2 1
250  4 2
235   H 2 1 20 90
240   H 1 3

16. Cp Q
260  1 3 2
250  2 4 C 4 60
235  H 2 3 4 1
240  H 1 3
THE COMPLETE MINIMUM UTILITY NETWORK

17. PINCH MATCH Pinch A Pinch Match Pinch 2 1 Exchanger 2 is not
Exchanger 2 is not
a pinch match
Pinch 1
Exchanger 3 is not
a pinch match

18. FEASIBILITY CRITERIA AT THE PINCH
Rule 1: Check the number of process streams and branches at the pinch
point
 Above the Pinch :
PINCH
PINCH
90
80
90
80 1 1 2 2 3 3 
(80+T1) 4 4
(80+T2) Q1 5 5 Q2
Tmin = 10C
Tmin = 10C

19. FEASIBILITY CRITERIA AT THE PINCH
Rule 1: Check the number of process streams and branches at the pinch
point
 Below the Pinch :
90
80
(90-T1)
90
80 1 1
(90-T2) 2 2  3 3 4 4 Q1 5 5 Q2
PINCH
PINCH
Tmin = 10C

20. FEASIBILITY CRITERIA AT THE PINCH
Rule 2: Ensure the CP inequality for individual matches are satisfied at the
pinch point.
 Above the Pinch :  Below the Pinch :
CPH1
CPC3 1 1
CPH2
CPC4 2 2 3 3 Q2 4 4
PINCH Q1
PINCH 1 T 2 T
Tmin
Tmin 3 4 Q Q Q2 Q1
CPC  CPH
CPC  CPH

21. Stream data at the pinch NH  NC? Yes No CPH  CPC Split a for every
pinch match
Split a
cold stream No
Yes
Split a
stream
( usually hot)
Place pinch
matches
Figure Design procedure above the pinch. (From B. Linnhoff et al., 1982.)

22. Stream data at the pinch NH  NC? Yes No CPH  CPC Split a for every
pinch match
Split a
cold stream No
Yes
Split a
stream
( usually hot)
Place pinch
matches
Figure Design procedure below the pinch. (From B. Linnhoff et al., 1982.)

23. CRITERION #3 THE CP DIFFERENCE
ABOVE THE PINCH,
INDIVIDUAL CP DIFFERENCE = CPC - CPH
OVERALL CP DIFFERENCE =
BELOW THE PINCH,
INDIVIDUAL CP DIFFERENCE = CPH - CPC
THE SUM OF THE INDIVIDUAL CP DIFFERENCES OF ALL PINCH
MATCHES MUST ALWAYS BE BOUNDED BY THE OVERALL CP
DIFFERENCE.

24. Overall CP Difference = 8 - 6 = 2
PINCH CP 4 2 5 3
Overall CP Difference = = 2
Total Exchanger CP Difference = = 2
O.K.

25. Overall CP Difference = 9 - 6 = 3
PINCH CP 4 2 5 3 1
Overall CP Difference = = 3
Total Exchanger CP Difference = = 2
O.K.

26. Overall CP Difference = 9 - 5 = 4
PINCH CP 3 2 8 1
Overall CP Difference = = 4
Total Exchanger CP Difference = = 6
Criterion violated !

27. Cp Q
260   1 3 2
250   130 2 4 C 4 60
235  180 135  H 2 3 4 1
240   H 1 3
Heat Load Loops
heat loads can be shifted around the loop from one unit
to another

28. 4 H 2 3 H 2 4 1 H C 1 3 C
Heat Load Loops
heat loads can be shifted around the loop from one unit
to another

29. 260   1 3 2
250  130 2 C 4 60
235    H 2 3 1
240   H 1 3
Heat Load Path
heat loads can be shifted along the path

30. 4 H 2 3 H 2 1 H C 1 3 C
Heat Load Path
heat loads can be shifted along the path

31. Cp Q
260   1 3 2 2
250   C 4
60+X
235  165  2 3 H 1
20+X
240   H 1 3
X=7.5

32. Two ways to break the loop
If:
L1>L4
L2>L3
then:
X=L4 or
X= -L3 1 1 2 2 3 4
(a) 3
L2 + X
L4 - X 4
L3 + X
L1 - X 1 2 3 2 1 4 3 4

33. heater/cooler can be included in a loop1 3 4 2
(b)
H1 - X 3 H
L3 + X 4 H
L4 - X
H2 + X 1 H 3 4 3 4
Figure Complex loops and paths

34. Match 1 is not in the path 1 2 (c) 3 4 H 1 2 4 3 C 2 3 1 4 C H 4 2 3
C + X 3
L3 + X
L4 - X 4 H
L2 - X
H + X H 1 2 4 2 3 4 3 C
Figure Complex loops and paths