ReflectionsonChrisFloudas andtheHeatExchangerNetwork
SynthesisProblemDr.AmyCiricMay6,2017
HotProcessStreamsReleaseHeat
ColdProcessStreamsAbsorbHeat
HeatExchangerNetwork
HotUtilitiesSupplyAdditional
Heat
ColdUtilitiesAbsorbExtra
Heat
FormalStatementoftheHeatExchangerNetworkSynthesis(HENS)Problem
A set of H hot streams with flow rates 𝑚"#(𝑖 = 1, . . , 𝐻) have to becooled from supply temperatures 𝑇".#(𝑖 = 1, . . , 𝐻) to targettemperatures 𝑇"/#(𝑖 = 1, . . , 𝐻). A set of C cold streams with flow rates𝑚01(𝑗 = 1, . . , 𝐶) have to be cooled from supply temperatures 𝑇0.1(𝑗 =1, . . , 𝐶) to target temperatures 𝑇0/1(𝑗 = 1, . . , 𝐶). It is required todetermine the structure of the heat exchanger network to achieve thisobjective at minimum total cost.
- Masso andRudd(1969)
EarlyWorkonHeatExchangerNetworkSynthesis
SecondLawLimitationstoHeatIntegration
450K
430K
360K
390K
350K
440K
420K
380K
350K
340K
QS=220kW
R1=0
R2=160kW
R4=350kW
R3=350kW
QW=350kW
PinchPoint
300
320
340
360
380
400
420
440
460
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300
T(K)
Q(kW)
HotComposite
ColdComposite
Chris’sFirstPublication:
Floudas,CiricandGrossmann(1986),AutomaticSynthesisofHeatExchangerNetworkConfigurations,AICHEJ.,32(2),276-290.
• IdentifiedpinchpointandminimumutilityconsumptionbyautomatingthePapoulias andGrossmann’s(1983)LPtransshipmentmodel.• IdentifiedstreammatchesandheatdutiesbyautomatingthePapouliasandGrossmann’s(1983)MILPtransshipmentmodel.• IdentifiedthenetworkstructurebysolvinganNLPbasedonChris’ssuperstructure.• Thiswasthefirstautomatedmethodforgeneratingoptimumheatexchangernetworks.
SuperstructureOptimization
Constructasuperstructurewithmanypossibledesignsembeddedwithinit..
Floudas,CiricandGrosssmann,1986
Constructasuperstructurewithmanypossibledesignsembeddedwithinit..
Series
Floudas,CiricandGrosssmann,1986
Constructasuperstructurewithmanypossibledesignsembeddedwithinit..
Parallel
Floudas,CiricandGrosssmann,1986
Constructasuperstructurewithmanypossibledesignsembeddedwithinit..
ParallelwithBypass
Floudas,CiricandGrosssmann,1986
Writeamodelofthesuperstructure..
4 𝑓678,#
�
1∈;<
= 𝐹#
𝑓18,# + 4 𝑓16
?,#�
6∈@<A
− 𝑓1C,# = 0
𝑓1E,# + 4 𝑓61
?,#�
6∈@<A
− 𝑓1C,# = 0
𝑇#𝑓18,# + 4 𝑡6
E,#𝑓16?,#
�
6∈@<A
− 𝑡18,#𝑓1
C,# = 0
𝑓1C,# 𝑡1
8,# − 𝑡1E,# = 𝑄#1
𝑓#C,1 𝑡#
E,1 − 𝑡#8,1 = 𝑄#1
∆𝑇1#1= 𝑡18,# − 𝑡#
E,1
∆𝑇2#1= 𝑡1E,# − 𝑡#
8,1
∆𝑇1#1≥ ∆𝑇K#L
∆𝑇2#1≥ ∆𝑇K#L
𝑘 ∈ 𝐻𝐶𝑇
𝑗 ∈ 𝑅#, 𝑘 ∈ 𝐻𝐶𝑇
𝑗 ∈ 𝑅#, 𝑘 ∈ 𝐻𝐶𝑇
𝑗 ∈ 𝑅#, 𝑘 ∈ 𝐻𝐶𝑇
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝐿𝑀𝑇𝐷#1 = 23S ∆𝑇1#1 T ∆𝑇2#1
U/W +∆𝑇1#1 + ∆𝑇2#1
2𝑖𝑗 ∈ 𝑀
..andusethatmodelasconstraintsinanoptimizationproblem.
min 4 𝑐𝑄#1
𝑈#1𝐿𝑀𝑇𝐷#1
].^�
#1∈_
4 𝑓678,#
�
1∈;<
= 𝐹#
𝑓18,# + 4 𝑓16
?,#�
6∈@<A
− 𝑓1C,# = 0
𝑓1E,# + 4 𝑓61
?,#�
6∈@<A
− 𝑓1C,# = 0
𝑇#𝑓18,# + 4 𝑡6
E,#𝑓16?,#
�
6∈@<A
− 𝑡18,#𝑓1
C,# = 0
𝑓1C,# 𝑡1
8,# − 𝑡1E,# = 𝑄#1
𝑓#C,1 𝑡#
E,1 − 𝑡#8,1 = 𝑄#1
∆𝑇1#1= 𝑡18,# − 𝑡#
E,1
∆𝑇2#1= 𝑡1E,# − 𝑡#
8,1
∆𝑇1#1≥ ∆𝑇K#L∆𝑇2#1≥ ∆𝑇K#L
𝑘 ∈ 𝐻𝐶𝑇
𝑗 ∈ 𝑅#, 𝑘 ∈ 𝐻𝐶𝑇
𝑗 ∈ 𝑅#, 𝑘 ∈ 𝐻𝐶𝑇
𝑗 ∈ 𝑅#, 𝑘 ∈ 𝐻𝐶𝑇
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝑖𝑗 ∈ 𝑀
𝐿𝑀𝑇𝐷#1 = 23S ∆𝑇1#1 T ∆𝑇2#1
U/W +∆𝑇1#1 + ∆𝑇2#1
2𝑖𝑗 ∈ 𝑀
Subjectto
ObjectiveFunction
Equationsdefiningthesearchspaceorfeasibleregion
..Solvingthisoptimizationproblemextractsthebestdesignfromthesuperstructure.
15kW/K
420⁰C
2.5kW/K7.5kW/K
7.5kW/K 2.5kW/K
Cost:$25,163
..Solvingthisoptimizationproblemextractsthebestdesignfromthesuperstructure.
15kW/K
420⁰C
5.3kW/K
Cost:$20,427
9.7kW/K
0kW/K
5.3kW/K
4.4kW/K
..Solvingthisoptimizationproblemextractsthebestdesignfromthesuperstructure.
15kW/K
420⁰C13.2kW/K
2.8kW/K
13.2kW/K
Cost:$19,850
..Solvingthisoptimizationproblemextractsthebestdesignfromthesuperstructure.
15kW/K
420⁰C15kW/K
Cost:$18,693
Ideally,allgoodsolutionsareinthesearchspace
Feasibleregion/searchspace infeasible
Feasibleregion/searchspace infeasible
Buthiddenconstraintscaneliminatesomegoodoptions.
Heuristicsanddecompositionapproachesartificiallylimitthesearchspace.
• Constantminimumtemperatureapproach• Nomatchesacrossthepinch• Minimizethenumberofmatchesbeforedesigningthenetwork• etc
Addyes/nodecisionstothesuperstructure
model
Strategytoreducethenumberofartificialconstraints:
MixedIntegerNonlinear
ProgrammingProblems(MINLP)
Chris’searlyworkatPrincetonappliedthisapproachtomanyprocesssynthesisproblems:• DistillationSequences• ReactorNetworks• HeatExchangerNetworks• PowerCycles
HeatExchangerNetworkSynthesisand
GlobalOptimization
HeatExchangerNetworkSynthesisandGlobalOptimizationEnergybalancesatthemixingpointsandacrosstheheatexchangershavebilinear(𝑥 T 𝑦) terms:
𝑇#𝑓18,# + 4 𝑡6
E,#𝑓16?,#
�
6∈@<A
− 𝑡18,#𝑓1
C,# = 0
𝑓1C,# 𝑡1
8,# − 𝑡1E,# = 𝑄#1
𝑓#C,1 𝑡#
E,1 − 𝑡#8,1 = 𝑄#1
..thesetermsarenonconvex andcreateanonconvexsearchspace:
feasible infeasible
Whichmayleadtomorethanonelocallyoptimalsolution.
infeasible𝑓 𝑥 =10
𝑓 𝑥 =5
𝑓 𝑥 =8
𝑓 𝑥 =3
feasible
HeatExchangerNetworkSynthesisandGlobalOptimization
Butifthetemperatureorflowratevariablesareheldconstant,theequationsbecomelinear:
𝑇#𝑓18,# + 4 𝑡6
E,#𝑓16?,#
�
6∈@<A
− 𝑡18,#𝑓1
C,# = 0
𝑓1C,# 𝑡1
8,# − 𝑡1E,# = 𝑄#1
𝑓#C,1 𝑡#
E,1 − 𝑡#8,1 = 𝑄#1
Chris’sfirstpaperonglobaloptimizationexploitedthisstructureoftheenergybalances
Floudas,AggarwalandCiric(1989):
• SolvethesuperstructureoptimizationproblemusingGeneralizedBendersDecomposition
• Choosetheflowrate-heatcapacityvariables𝒇 asthecomplicatingvariables
Convexmasterproblemprovides• Valuesof𝒇• Lowerboundontheglobal
optimum
Convex primalproblemprovides• Valuesof𝒕• Upperboundontheglobal
optimum
Successiveiterationsbetweenthemasterandprimalproblemsconvergetotheglobaloptimum
PersonalReflections
𝐼𝑛𝑠𝑖𝑔ℎ𝑡 T 𝑅𝑖𝑔𝑜𝑟 T 𝐸𝑛𝑒𝑟𝑔𝑦 T 𝐸𝑛𝑡ℎ𝑢𝑠𝑖𝑎𝑠𝑚
𝑇ℎ𝑎𝑛𝑘𝑦𝑜𝑢!
1959-2016