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