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Heat Exchange Network Synthesis (H (HENS) Stage-wise Superstructure for Synthesis of Heat Exchange Networks
Cheng-Liang Chen
PSE
Given:
A set of hot process streams to be cooled, and a set of cold process streams to be heated ☞ Available heating/cooling utilities ☞ Inlet/outlet temperatures and heat capacity flow rates for all streams and utilities ☞ Area cost and fixed unit cost, utility costs ☞
➢
Determine: Network Configurations to
Minimize the total annual cost (TAC) ☞ Maximize operating flexibility (operating ranges of T & F ) ☞
LABORATORY
Department of Chemical Engineering National TAIWAN University
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Heat Exchange Network Synthesis (H (HENS)
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Stage-wise Superstructure for HENS All Possible Matches; Isothermal Mixing
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Stage-wise Superstructure for HENS
Stage-wise Superstructure for HENS
Order of Matches; Isothermal Mixing
Utilities for Heating/Cooling; Isothermal Mixing
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Modeling the Stage-wise Superstructure Indices, Sets, Parameters
i
hot process stream
j
cold process stream
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Modeling the Stage-wise Superstructure Continuous and Binary Variables
Positive variables
Indices and sets
k
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stage
hu
hot utility
cu
cold utility
in
inlet
out
outlet
aijk , ai,cu, ahu,j tik tjk thijk tcijk dtijk dtout i,cu dtout hu,j
areas for exchangers temperature of hot stream i at the temperature location k temperature of hot stream j at the temperature location k temperature for part of hot i that is connected to cold j in stage k temperature for part of cold j that is connected to hot i in stage k temperature difference for match (ij) at the temperature location k temperature difference for match (i,cu) at the hot end of the heat exchanger temperature difference for match (ij) at the hot end of the heat exchanger temperature difference for match (ij) at the cold end of the heat exchanger heat exchanged between hot steam i and cold steam j in stage k heat exchanged between hot steam i and cold utility heat exchanged between hot utility and cold steam j split ratio of hot i that is connected to cold j in stage k split ratio of cold j that is connected to hot i in stage k
T iin , T iout, T jin , T jout
inlet and outlet temperatures
EMAT
minimum-approach temperature difference, ∆T min
F C i , F C j
heat capacity flowrates
U ij , U i,cu, U hu,j
overall heat transfer coefficients
Ccu , Chu
per unit cost of cold and hot utility
dthijk dtcijk qijk qi,cu qhu,j rhijk rcijk
CFij , CFi,cu , CFhu,j
fixed charges for exchangers
Binary variables
CAij , CAi,cu , CAhu,j
area cost coefficients
NOK
total number of stages
Λ
upper bound for heat exchange
zijk zi,cu zhu,j
Γ
upper bound for temperature difference
Parameters
temperature difference for match ( hu,j) at the cold end of the heat exchanger
existence of a unit for the match (ij) in stage k existence of a unit for the match (i,cu) existence of a unit for the match ( hu,j)
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Modeling the Stage-wise Superstructure
Modeling the Stage-wise Superstructure
Overall Heat Balance for Hot Streams
Overall Heat Balance for Cold Streams
q q
ijk + q i,cu
j ∈CP k∈ST
ijk + q hu,j
i∈HP k∈ST
= F C i T iin − T iout =
F C T j
out j
− T in j
ijk + q i,cu
= F C i T iin − T iout
ijk + q hu,j
=
q q
∀i ∈ HP (1)
j ∈CP k∈ST
∀ j ∈ CP (2)
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i∈HP k∈ST
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F C T j
out j
− T in j
∀i ∈ HP (1) ∀ j ∈ CP (2)
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Modeling the Stage-wise Superstructure
Modeling the Stage-wise Superstructure
Stage-wise Heat Balance for Hot Streams
Stage-wise Heat Balance for Cold Streams
q q
ijk
= F C i (tik − ti,k+1)
∀i ∈ HP, ∀k ∈ ST (3)
j ∈CP
ijk
i∈HP
q q
ijk
= F C i (tik − ti,k+1)
∀i ∈ HP, ∀k ∈ ST (3)
ijk
= F C j (tjk − tj,k+1) ∀ j ∈ CP , ∀k ∈ ST (4)
j ∈CP
= F C j (tjk − tj,k+1) ∀ j ∈ CP, ∀k ∈ ST (4)
i∈HP
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Modeling the Stage-wise Superstructure
Modeling the Stage-wise Superstructure
Hot and Cold Utility Loads
Assignment of I/O Temperatures and Feasibility of Temperatures
q hu,j = F C j T jout − tj1
q i,cu
tj,NOK+1
= F C i (ti,NOK+1 −
∀ j ∈ CP (5)
T iout)
dtin i,cu dtin hu,j
∀i ∈ HP (6)
tik ti,NOK+1 tjk T jout
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T iin
= = = = ≥ ≥ ≥ ≥
ti1
T jin in T iout − T cu in T hu − T jout
ti,k+1 T iout tj,k+1 tj1
∀i ∈ HP ∀ j ∈ CP ∀i ∈ HP ∀ j ∈ CP HP , ∀k ∈ ST ∀i ∈ ∀i ∈ HP ∀ j ∈ CP, ∀k ∈ ST ∀ j ∈ CP
(7) (8) (9) (10) (11) (12) (13) (14)
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Modeling the Stage-wise Superstructure Other Constraints ➢
q i,cu q hu,j dtijk dtij,k+1 dtout i,cu dtout hu,j
≤ ≤ ≤ ≤ ≤ ≤ ≤
Λzijk Λzi,cu Λzhu,j tik − tjk + Γ(1 − zijk ) ti,k+1 − tj,k+1 + Γ(1 − zijk ) out + Γ(1 − zi,cu) ti,NOK+1 − T cu out T hu − tj1 + Γ(1 − zhu,j )
T jout
∀i ∈ HP , ∀ j ∈ CP , ∀k ∈ ST ∀i ∈ HP ∀ j ∈ CP ∀i ∈ HP, ∀ j ∈ CP , ∀k ∈ ST ∀i ∈ HP , ∀ j ∈ CP , ∀k ∈ ST ∀i ∈ HP ∀ j ∈ CP
Minimum approach-temperatures dtijk dtout i,cu out dt
Variable bounds T iin
Logical constraints q ijk
➢
➢
≥ EMAT ∀i ∈ HP , ∀ j ∈ CP, ∀k ∈ ST ∪ {NOK + 1 } (22)∗ ≥ EMAT ∀i ∈ HP (23) EMAT ∀ CP (24)
(15) (16) (17) (18)∗ (19)∗ (20) (21)
q ijk q ijk q i,cu q hu,j
≥ tik ≥ T iout ≥ tjk ≥ T jin ≤ ≤ ≤ ≤
∀i ∈ HP , ∀k ∈ ST ∀ j ∈ CP , ∀k ∈ ST
F C i T iin − T iout
F C T F C T F C T j i
j
− T − T − T
out j in i out j
in j out i in j
∀i ∈ HP , ∀ j ∈ CP, ∀k ∈ ST ∀i ∈ HP , ∀ j ∈ CP, ∀k ∈ ST ∀i ∈ HP ∀ j ∈ CP
(25) (26) (27) (28) (29) (30)
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Objective Function: TAC
MINLP Formulation: Isothermal Mixing
i∈HP j ∈CP k∈ST
ij
ijk )
βijk
+
i∈HP
=
LMTDijk
=
≈
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Ω
=
dtijk dtij,k+1
x
j ∈CP
x
(31)
≡
zijk , zi,cu , zhu,j ; out tik , tjk ; dtijk , dtout i,cu, dthu,j ;
q ijk , q i,cu , q hu,j
∀i ∈ HP , ∀ j ∈ CP, ∀k ∈ ST
(33)
dtijk +dtij,k+1 2
=
qijk
=
qi,cu qhu,j
= =
i∈HP
= =
in
1/3
(34) (Chen Approximation)
out in F C j T j − T j F C i tik − ti,k+1 F C j tjk − tj,k+1 out F C i ti,NOK+1 −T i out in out F C i T i − T i
F C j T j
− tj 1
out
ti1 = T i
tj,NOK+1 = T j
in out in dti,cu = T i − T cu
≤
hu,j ( ahu,j )
(32)
qijk + qi,cu j ∈CP k∈ST qijk + qhu,j i∈HP k∈ST qijk j ∈CP
qijk
x∈Ω
βhu,j
j ∈CP
dtijk − dtij,k+1 dtijk ln dtij,k+1
+
hu hu,j
q ijk U ij · LMTDijk
βi,cu
i∈HP
+
cu i,cu
aijk
hu,j zhu,j
j ∈CP
i,cu)
i,cu
min TAC
CF CA
+
i,cu i,cu
i∈HP
i∈HP j ∈CP k∈ST
+
CF z CA (a C q
+
ij ijk
+
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Modeling the Stage-wise Superstructure
CF z CA (a C q
TAC =
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Modeling the Stage-wise Superstructure
in in out dthu,j = T hu − T j
tik ≥ ti,k+1
out ti,NOK+1 ≥ T i
tjk ≥ tj,k+1
T j
out
≥ tj 1
Λzijk , qi,cu
≤
Λzi,cu, qhu,j
dtijk
≤
tik − tjk + Γ(1 − zijk )
dtij,k+1
≤
ti,k+1 − tj,k+1 + Γ(1 − zijk )
out dti,cu out dthu,j
≤ ≤
∗ dtijk ≥ EMAT ,
in
T i
out
T j
≥ ≥
qijk , qi,cu
≤
qijk , qhu,j
≤
≤
Λzhu,j ∗
∗
out ti,NOK+1 − T cu + Γ 1 − zi,cu out T hu − tj 1 + Γ 1 − zhu,j out out dti,cu , dthu,j ≥ EMAT out tik ≥ T i in tjk ≥ T j in out F C i T i − T i out in − F C j T j T j
∀i ∈ HP , ∀j ∈ CP , ∀k ∈ ST
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Simultaneous Optimization Model for Heat Exchanger Network Synthesis One Problem with 2-Hot-2-Cold Streams Stream
T in
T out
F Cp (kW/K)
H 1
650
370
10.0
1.0
-
H 2
590
370
20.0
1.0
-
C 1
410
650
15.0
1.0
-
C 2
353
500
13.0
1.0
-
S 1
680
680
−
5.0
80
W 1
300
320
−
1.0
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Assume ∆ T min = 10K ,
h (KW/m2K) Cost ($/KW-yr)
Exchanger cost = $5500 + 150 A (area, m 2)
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HENS: Simultaneous Optimization
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Simultaneous MINLP Model: Example
Optimal Network Structure
H1 C1 C2 S1 W1
155, 000/yr total cost (71, 400 for utility cost and 83, 600 for capital cost) Chen CL
F Cp (kW/K)
T in (K)
T out (K)
h (kW/m2K)
Cost ($/kW-yr)
22. 20. 7.5 − −
440 349 320 500 300
350 430 368 500 320
2.0 2.0 .67 1.0 1.0
120 20
Min recovery app temp = 1 K Exchanger Cost = 6, 600 + 670(area)0.83
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Simultaneous MINLP Model: Same Example with No Stream Splitting
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Thank You for Your Attention Questions Are Welcome
