Supplement I: Influences of Tmin on the hot utility demand
Theorem 1 requires that the compression work can be completely converted into
savings in hot utility when Pinch Compression is implemented. This condition was
satisfied by constraining the maximum portion using Pinch Compression with the
concept of Potential Pinch Points. The Grand Composite Curve (GCC) has been used as a
tool in a systematic design procedure. However, the identity of streams (hot/cold) to be
compressed and the relative position of stream temperatures (supply/target) were not
considered. The corresponding effects are illustrated in this Appendix.
The temperatures of interest are the supply ( Ts ) and target ( Tt ) temperatures of the
stream to be compressed, the pinch temperature ( TPI ), and the outlet temperature of Pinch
Compression ( Tcomp ,PI ). With 4 different temperatures for a stream there are (4!)  24
distinctly different sequences of these temperatures. Since the relative position of Ts and
Tt is given by the stream identity, and Tcomp ,PI  TPI , the total number of cases that needs
to be investigated is reduced to 24 / (2  2)  6 for cold streams and the same number for
hot streams. All the 12 cases are listed in Table S1, from which 4 cases are selected for
detailed analyses illustrating hot or cold streams with supply temperature above or below
pinch: C.1 (cold stream to be compressed and Ts  TPI ), C.4 (cold stream to be
compressed and Ts  TPI ), H.1 (hot stream to be compressed and Ts  TPI ) and H.2 (hot
stream to be compressed and Ts  TPI ).
Tcomp,PI
TPI  Tmin
Tcomp,PI
Tcomp,PI
TPI  Tmin
TPI
TPI
(a)
Tt
Tt
TPI
(b)
TPI
Tt
Tt
Ts
Ts
TPI  Tmin
Tcomp,PI
Tt
(c)
Tcomp,PI
Tcomp,PI
TPI  Tmin
(d)
Tt
Ts
Ts
TPI
TPI
TPI
Tcomp,PI
TPI
Tcomp,PI
Tcomp,PI
Tt
Tt
(e) T  T
PI
min
Tcomp,PI
Ts
(f) T  T
PI
min
Ts
TPI
TPI
TPI
Tcomp,PI
TPI
TPI
Tcomp,PI
Tcomp,PI
Tcomp,PI
Ts
TPI
(h)
(g)
TPI  Tmin
Ts
Ts
TPI
TPI
Tt
Tt
Figure S1. Illustration of temperature change for the stream being compressed in
various Cases: (a) C.1.i, (b) C.1.ii, (c) C.4.i, (d) C.4.ii, (e) C.4.iii, (f) C.4.iv, (g) H.1.i,
(h) H.2.
Table S1. Underestimation of hot utility for Pinch Compression
Cases
Sub-cases
Underestimation of hot utility
C.1: Ts  Tt  TPI  Tcomp ,PI
(i): TPI  Tmin  Tcomp,PI
mc p Tmin
(ii) : Tcomp,PI  TPI  Tmin
mc p Tmin
(i): TPI  Tmin  Tt
0
(ii): Tt  TPI  Tmin  Tcomp,PI
ymc p Tmin , 0  y  1
(iii): Tt  Tcomp,PI  TPI  Tmin
ymc p Tmin , 0  y  1
C.3: Ts  TPI  Tcomp ,PI  Tt
-
0
C.4: TPI  Ts  Tt  Tcomp ,PI
(i): Tcomp,PI  TPI  Tmin
ymc p Tmin , 0  y  1
(ii): Tt  TPI  Tmin  Tcomp,PI
ymc p Tmin , 0  y  1
(iii): TPI  Tmin  Ts
mc p Tmin
(iv): Ts  TPI  Tmin  Tt
ymc p Tmin , 0  y  1
(i) TPI  Tmin  Ts
mc p Tmin
(ii) TPI  Tmin  Ts
ymc p Tmin , 0  y  1
(i) TPI  Tmin  Ts
mc p Tmin
(ii) Ts  TPI  Tmin
ymc p Tmin , 0  y  1
(i) Ts  TPI  Tmin
mc p Tmin
(ii) TPI  Tmin  Ts
ymc p Tmin , 0  y  1
H.2: Tt  TPI  Ts  Tcomp ,PI
-
0
H.3: Tt  TPI  Tcomp ,PI  Ts
-
0
H.4: TPI  Tt  Ts  Tcomp ,PI
-
0
H.5: TPI  Tt  Tcomp ,PI  Ts
-
0
H.6: TPI  Tcomp ,PI  Tt  Ts
-
0
C.2: Ts  TPI  Tt  Tcomp ,PI
C.5: TPI  Ts  Tcomp ,PI  Tt
C.6: TPI  Tcomp ,PI  Ts  Tt
H.1: Tt  Ts  TPI  Tcomp ,PI
Case C.1: The stream is compressed after being heated from Ts to TPI , and then cooled
from Tcomp ,PI to Tt . The case is studied in two sub-cases: TPI  Tmin  Tcomp,PI (Case C.1.i)
and Tcomp,PI  TPI  Tmin (Case C.1.ii), as shown in Figure S1(a-b). For Case C.1.i, the
heating from Ts to Tt (before Pinch Compression) is part of the original GCC used to
determine QHU ,0 , and the heating from Tt to TPI can be satisfied by the cooling of the
stream from TPI  Tmin to Tt  Tmin after compression. After Pinch Compression, the
heat from the cooling of the stream from Tcomp ,PI to TPI  Tmin can be used to heat other
streams. The hot utility demand is thus reduced by mc p (T comp,PI TPI )  mc p Tmin , indicating
that the compression work can not be completely converted into heat savings. The
difference is mc p Tmin , thus the heating demand is underestimated by an amount of
mc p Tmin . For Case C.1.ii, the compression heat can not be utilized to heat other streams
since Tcomp,PI  TPI  Tmin , in addition, the heating of the stream from Tt to TPI can not be
completely satisfied by recuperative heating. The total effect is that the hot utility is
underestimated by mc p Tmin .
Case C.4: The stream is compressed after being cooled from Ts to TPI , and then
cooled from Tcomp ,PI to Tt . As shown in Figure S1(c-f), the following four sub-cases are
studied: Tcomp,PI  TPI  Tmin (Case C.4.i), Tt  TPI  Tmin  Tcomp,PI (Case C.4.ii),
TPI  Tmin  Ts (Case C.4.iii) and Ts  TPI  Tmin  Tt (Case C.4.iv). For Case C.4.i, the
cooling from Ts to TPI before compression and from Tcomp ,PI to Tt after compression can
not be utilized due to the limitation of Tmin (both Tcomp ,PI and Ts are less than TPI  Tmin ).
However, the heating of the stream from Ts to Tt is no longer required when Pinch
Compression is used. The target on savings in hot utility by using Pinch Compression is
equal to the compression work, i.e., mc p (T comp,PI TPI ) . The actual savings is mc p (Tt  Ts ) .
The underestimation of hot utility is thus equal to mc p (T comp,PI TPI )  mc p (Tt  Ts ) . Since
the value is less than mc p Tmin , it can be written as ymc p Tmin where 0  y  1 . For Case
C.4.ii, the cooling from Ts to TPI before compression and from TPI  Tmin to Tt after
compression can not be utilized, and the underestimation of hot utility is thus ymc p Tmin
where 0  y  1 . For Case C.4.iii, the cooling from TPI  Tmin to TPI before compression
can not be utilized, and the underestimation of hot utility is thus mc p Tmin . For Case.4.iv,
after compression, the cooling from Tcomp ,PI to Tt can be utilized and thus reduce the
heating demand. The original heating from Ts to Tt (before compression) is no longer
required. However, the cooling from Ts to TPI before compression can not be utilized.
The net effect is that a portion of the compression work can not be completely converted
into heat saving, i.e., there is an underestimation of hot utility by ymc p Tmin , 0  y  1 .
Case H.1: The stream is compressed after being heated from Ts to TPI , and then
cooled from Tcomp ,PI to Tt . The following two sub-cases are analyzed: Ts  TPI  Tmin
(Case H.1.i) and TPI  Tmin  Ts (Case H.1.ii). For Case H.1.i, as shown in Figure S1(g),
the heating of the stream from Ts to TPI  Tmin before compression can be satisfied by the
cooling of the stream from TPI to Ts  Tmin (recuperative heating). However, the heating
from TPI  Tmin to TPI increases the heating demand by mc p Tmin . After compression, the
cooling from Tcomp ,PI to TPI can be completely converted into heat savings. The
underestimation of hot utility is thus mc p Tmin . For Case H.1.ii, recuperative heating can
not be used for the heating from Ts to TPI , thus the underestimation of hot utility is
ymc p Tmin , where 0  y  1 .
Case H.2: The stream is compressed after being cooled from Ts to TPI , and then
cooled from Tcomp ,PI to Tt . This case is shown in Figure S1(h). The cooling from Ts to TPI
before compression and the cooling from TPI to Tt after compression is part of the
original GCC used to determine QHU ,0 . The net effect is that the heating demand is
reduced by mc p (T comp,PI TPI ) . The compression work is thus completely converted into
heat savings and there is no underestimation of hot utility.
Similar analyses can be performed for the other cases. The results are summarized in
Table S1. It can be concluded that the heating demand is underestimated by an amount of
ymc p Tmin where 0  y  1 , when Pinch Compression is used following the design
procedure presented in this paper. However, the advantage of neglecting the issues of
stream identity and relative positions of Ts and Tt is that the traditional GCC can be used.
Supplement II: Proof of Theorem 2
Theorem 2 is proven in the following way. A cold stream is assumed to be compressed
from p s to pt . In the case that a hot stream is compressed, a similar proof can be
established. According to condition (2) in Theorem 2, the heating demand can be
completely satisfied by Pinch Compression. The outlet temperature of Pinch
Compression, Tcomp ,PI , should thus be higher than the lowest possible hot utility
temperature, THU ,min , as shown in Figure S2 (modified temperatures are used). In the proof
of Theorem 1, it was explained that Pinch Compression is more favorable than
compression schemes above pinch. The comparison is thus performed between Pinch
Compression and below pinch compression. The following two cases are compared: Case
A - compression starts at temperature TA , T0  TA  TPI , and the heating demand is
satisfied after the compression heat is included; Case B - the stream ( mc p ) to be
compressed is split into two portions: one portion (α) is compressed at TPI and the
compression work is equal to the heating demand QHU ,0 , while the remaining portion (γ)
is compressed at T0 . Case A actually includes two sub-cases: the first case (A.i) is that the
heating demand is satisfied and Tcomp , A  THU ,min (the stream is not split); the second case
(A.ii) is that the compression heat (above pinch) is more than required and the stream is
split: the first portion is compressed at TA and the heating demand is satisfied, and the
remaining portion is compressed at T0 .
T’ (oC)
QHU,0
T’HU, min
T’comp,PI
T’comp,A
QCU,0 : cold utility demand for the case without pressure
manipulation
QHU,0 : hot utility demand for the case without pressure
manipulation
T’A : inlet temperature for compression in Case A
T’B : inlet temperature for compression in Case B
T’PI
T’comp,A : outlet temperature for compression in Case A
T’B
T’comp,PI : outlet temperature for Pinch Compression
T’A
T’HU,min : the lowest possible hot utility temperature
QCU,0
T’PI : pinch temperature
T’0
T’0 : ambient temperature
H (kW)
Figure S2. GCC without pressure manipulation for Theorem 2.
Since the heating demand is completely satisfied by the compression heat in Cases A
and B, the comparison can thus be performed on work consumption. The first comparison
is performed between Case A.i (no portion is compressed at T0 ) and Case B. For Case A.i,
assuming that the compression ratios below and above pinch are pr , A1 and pr , A2
respectively, obviously pr , A1 pr , A2  pr  pt / ps . The work consumption for Case A.i is
WA  mc p (Tcomp, A  TA )  mc p (Tcomp , A  TPI )  mc p (TPI  TA )  QHU ,0  mc p (TPI  TA ) . Similarly for
Case B, WB  (mc p ) (Tcomp,PI  TPI )  (mc p ) (Tcomp ,0  T0 )  QHU ,0  (mc p ) (Tcomp,0  T0 ) . The
following mc p ratios can be derived from the relations above since (mc p )  (mc p )  mc p :
(mc p )
mc p
(mc p )
mc p


Tcomp , A  TPI
Tcomp ,PI  TPI

mc p  (mc p )
mc p
TPI [ pr , A2( nc 1)/ nc  1] pr , A2( nc 1)/ nc  1

TPI [ pr ( nc 1)/ nc  1]
pr ( nc 1)/ nc  1

pr ( nc 1)/ nc  pr , A2( nc 1)/ nc
pr ( nc 1)/ nc  1
The work consumption for the two cases can thus be compared:
WA  WB  mc p (TPI  TA )  (mc p ) (Tcomp ,0  T0 )
 mc p {TA [ pr , A1( nc 1)/ nc  1] 
( mc p )
mc p
T0 [ pr ( nc 1)/ nc  1]}
 mc p [ pr , A1( nc 1)/ nc  1][TA  T0 pr , A2 ( nc 1)/ nc ]
 mc p [ pr , A1( nc 1)/ nc  1][1 / pr , A1( nc 1)/ nc ][TA pr , A1( nc 1)/ nc  T0 pr ( nc 1) / nc ]
 mc p [ pr , A1( nc 1)/ nc  1][1 / pr , A1( nc 1)/ nc ](TPI  Tcomp ,0 )
( n 1)/ n
 1  0 and TPI  Tcomp ,0 according to condition (1) in Theorem 2, it is
Since pr , A1
c
c
concluded that WA  WB  0 , i.e., Case B consumes less work.
For Case A.ii, one portion is compressed at T0 . This portion can be removed from the
comparison by subtracting an equal portion with Ambient Compression in Case B. The
previous proof can then be applied and the same conclusion is achieved. There may also
be cases where the stream is split into many portions that are compressed at different
temperatures below pinch, however, the compression of each portion consumes more
exergy compared to the case where Pinch Compression combined with Ambient
Compression (if necessary) are used for the same portion (see the comparison between
Cases A.i and B). The total effect is that Case B has the minimum exergy consumption.
Theorem 2 has thus been proven.
Supplement III: Proof of Theorem 3
Figure S3 shows possible compression alternatives in the GCC: the first portion (α) is
compressed at T ' PI , the second portion (β) is compressed at T 'comp ,0 , and the third portion
(γ) is compressed at T '0 . The following two cases exist and are discussed.
T’ (oC)
QHU,0
T’comp,PI
Qα
Qβ
Qγ
T’HU, min
T’comp,0
QT’comp,0
T’PI
QCU,0
T’0
H (kW)
Figure S3. GCC for Theorem 3.
Case (1): mc p (T 'comp ,0  T ' PI )  QT '
Here, QT '
comp ,0
comp ,0
is the heating demand at T 'comp ,0 , no new pinch is created at T 'comp ,0 . The
heating demand is satisfied by a combination of Pinch Compression and Ambient
Compression, i.e., the portion β in Figure S3 is not required. In line with Theorem 1, any
compression scheme above pinch consumes more exergy than Pinch Compression.
The compression scheme proposed above (referred to as Case B) is compared with
below pinch compression (Case A) in the following way. For Case A, assume that
compression starts at TA  TPI and the heating demand is satisfied by the compression heat.
The compression ratios below and above pinch are pr , A1 and pr , A2 respectively,
pr , A1 pr , A2  pr  pt / ps , TPI  TA pr , A1( nc 1)/ nc and Tcomp , A  TA pr ( nc 1)/ nc . The work consumption
for Case A is WA  mc p (Tcomp , A  TA ) . The heating demand is satisfied by the compression
heat, thus QHU ,0  mc p (Tcomp , A  TPI ) .
For the portion γ in Case B (Pinch Compression combined with Ambient
Compression), the compression ratios below and above pinch are pr ,B1 and pr ,B 2
( n 1)/ n
( n 1)/ n
respectively, pr ,B1 pr ,B 2  pr  pt / ps , TPI  T0 pr ,B1
and Tcomp ,0  T0 pr
. The work
c
c
c
c
consumption is WB  (mc p ) (Tcomp,PI  TPI )  (mc p ) (Tcomp ,0  T0 ) . The heating demand is
satisfied by the compression heat, thus QHU ,0  (mc p ) (Tcomp ,PI  TPI )  (mc p ) (Tcomp ,0  TPI ) .
The heating demand is the same for both cases, i.e., QHU ,0  mc p (Tcomp , A  TPI )
 (mc p ) (Tcomp ,PI  TPI )  (mc p ) (Tcomp ,0  TPI ) , which can be written as:
[( mc p )  ( mc p ) ]TPI ( pr , A2 ( nc 1)/ nc  1)  ( mc p ) TPI ( pr ( nc 1)/ nc  1)  ( mc p ) TPI ( pr ,B 2( nc 1)/ nc  1) . The
following mc p ratios can be derived:
(mc p )
(mc p )
(mc p )
mc p

pr ( nc 1)/ nc  pr , A2( nc 1)/ nc
pr , A2( nc 1)/ nc  pr ,B 2( nc 1)/ nc

pr ( nc 1)/ nc  pr , A2( nc 1)/ nc
pr ( nc 1)/ nc  pr ,B 2( nc 1)/ nc
Since the heating demand is satisfied in both cases, only the work consumption is
compared:
WA  WB
 mc p (Tcomp , A  TA )  [( mc p ) (Tcomp ,PI  TPI )  ( mc p ) (Tcomp ,0  T0 )]
 [QHU ,0  mc p (TPI  TA )]  [QHU ,0  ( mc p ) (TPI  T0 )]
 mc p (TPI  TA )  ( mc p ) (TPI  T0 )
 mc p {TA [ p
 mc p [ p


( nc 1)/ nc
r , A1
( nc 1)/ nc
r , A1
pr ( nc 1)/ nc  pr , A2 ( nc 1)/ nc
 1]  ( nc 1)/ nc
T [ pr , B1( nc 1)/ nc  1]}
( nc 1)/ nc 0
pr
 pr , B 2
pr , A2 ( nc 1)/ nc
 1][TA  T0
]
pr , B 2 ( nc 1)/ nc
mc p [ pr , A1( nc 1)/ nc  1]
pr , A1( nc 1)/ nc
mc p [ pr , A1( nc 1)/ nc  1]
pr , A1( nc 1)/ nc
[TA pr , A1( nc 1)/ nc  T0 pr ,B1( nc 1)/ nc ]
(TPI  TPI )
0
Thus, the exergy consumption is the same for Cases A and B. Since Pinch
Compression for the entire stream produces more heat than required, the stream can be
split into two portions which are compressed at T0 and TPI respectively (Case B).
Alternatively, the entire stream can be compressed below pinch temperature ( T0  TA  TPI )
without splitting (Case A). It is reasonable that the two cases have the same work (exergy)
consumption, since the compression in Case A is performed at an intermediate
temperature between the two inlet temperatures ( T0 and TPI ) for compression of the two
branches in Case B.
The below pinch compression includes other cases where the compression heat (above
pinch) is more than required and the stream needs to be split: the first portion is
compressed at TA ( T0  TA  TPI ) and the remaining portion is compressed at T0 in such a
way that the heating demand is satisfied by the compression heat. The portion with
Ambient Compression can be removed from the comparison by subtracting an equal
portion with Ambient Compression in Case B. The previous proof can then be applied
and the same conclusion can be achieved. Similar to the proof of Theorem 2, the below
pinch compression may also include cases where the stream is split into many portions
that are compressed at different temperatures below pinch. The compression of each
portion can be compared with a corresponding case where Pinch Compression combined
with Ambient Compression (if necessary) is used for the same portion. The total effect is
that the two cases (A and B) have the same amount of exergy consumption.
In conclusion, the compression scheme proposed consumes the smallest amount of
exergy. It should be noted that some cases with compression at an intermediate
temperature ( T0  T  TPI ) can achieve the same minimum exergy consumption, and the
capital cost may even be lower since the number of stream splits can be reduced, however,
the determination of the inlet temperatures of compression is not graphically
straightforward for complex cases where new pinches are created when the compression
heat is integrated, as illustrated by the design procedure presented in Figure 9 and
illustrated by Example 5. The objective of this paper is to develop a straightforward
design methodology that achieves the target of minimum exergy consumption. The
consideration of capital cost as well as retrofit of HENs will be investigated in future
work.
Case (2): mc p (T 'comp ,0  T ' PI )  QT '
comp ,0
Ambient Compression creates a new pinch point at T 'comp ,0 . The maximum portion that
can be compressed at T ' PI is constrained by the new pinch point. The heat from Pinch
Compression above T 'comp ,0 has to be used above the new pinch ( T 'comp ,0 ). If the heating
demand above T 'comp ,0 can not be satisfied by Pinch Compression, a portion (β) is
compressed at the new pinch ( T 'comp ,0 ). The three portions can be determined by the
following equations:
(mc p ) (T 'comp ,0  T ' PI )  ( mc p ) (T 'comp ,0  T ' PI )  QT 'comp ,0
(mc p ) (T 'comp ,PI  T ' PI )  ( mc p )  (T 'comp ,Tcomp ,0  T 'comp ,0 )  ( mc p ) (T 'comp ,0  T ' PI )  QHU ,0
(mc p )  (mc p )   (mc p )  mc p
Similar to the proof for Case (1), it can be proven that the combination of Pinch
Compression (portion α) and Ambient Compression (portion γ) consumes the minimum
amount of exergy when the heating demand is QHU ,0  Q (accounting for the heating
effect from compression of portion β). Since a new pinch is created at T 'comp ,0 , according
to Theorem 1, the heating demand Q should be satisfied by compression at the new
pinch (portion β). The proposed scheme thus has the minimum exergy consumption.
If (mc p ) is too large and QT '
comp ,0
can be completely satisfied by Ambient Compression,
no feasible solution can be found for the above equations, instead the three portions can
be determined by the following equations:
(mc p )  0
(mc p )  (T 'comp ,Tcomp ,0  T 'comp ,0 )  QT 'comp ,0  QHU ,0
(mc p )  (mc p )   (mc p )  mc p
Since a new pinch is created at T 'comp ,0 , according to Theorem 2, a combination of the
(new) Pinch Compression and Ambient Compression has the minimum exergy
consumption.
Similar to Case (1), some cases where compression at some intermediate temperature
( T0  T  TPI ) is used in combination with Ambient Compression (if necessary) may
achieve the same minimum exergy consumption.