mathematical programming model for heat-exchanger network synthesis including detailed...

Mathematical Programming Model for Heat-Exchanger NetworkSynthesis Including Detailed Heat-Exchanger Designs. 2. NetworkSynthesis

Fabio T. Mizutani,† Fernando L. P. Pessoa,† and Eduardo M. Queiroz†

Departamento de Engenharia Quı́mica, Escola de Quı́mica, Universidade Federal do Rio de Janeiro,Caixa Postal 68542, Rio de Janeiro, RJ, 21949-900 Brazil

Steinar Hauan and Ignacio E. Grossmann*

Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

This work proposes an optimization model for heat-exchanger network synthesis that includesa heat-exchanger design model. This model takes into account several detailed designvariables: number of tubes, number of tube passes, internal and external tube diameters, tubearrangement pattern, number of baffles, head type, and fluid allocation (i.e., to the shell ortubes). The network superstructure with individual heat-exchanger designs is solved using thelogic-based outer approximation method (Turkay, M.; Grossmann, I. E. Comput. Chem. Eng.1996, 20, 959-978). An interesting feature of the model is that it contains disjunctions fortopology selection, which in turn has disjunctions for the heat-exchanger design. The proposedmodel determines the heat-exchanger network that minimizes the total annualized costaccounting for area, pumping, and utility expenses. Examples are presented to illustrate thismethod.

Introduction

Heat-exchanger network synthesis (HENS) has beenthe subject of a significant research over the last 40years.2 Several papers in the chemical process synthesisarea have developed mathematical programming toolsto solve this problem. However, most formulationsassume constant heat-transfer coefficients and coun-terflow arrangement for all stream matches, which canlead to nonoptimal results because they usually are farfrom the operational reality because heat-transfer coef-ficients are strongly influenced by the exchanger geom-etry and multipass shell-and-tube units are frequentlyfound in chemical process plants.

Gundersen and Naess3 presented an extensive HENSreview, and recently Furman and Sahinidis2 reportedthat over 400 papers have been published on the subjectover the last 4 decades. The minimum utility target wasproperly defined in the literature because Linnhoff andHindmarsh4 proposed the energy cascade algorithm inthe well-known pinch-point method. At the same time,several mathematical programming methods have beenproposed to solve the problem. Papoulias and Gross-mann5,7 tackled the minimum energy usage and theminimum number of units. Floudas et al.6 formulateda superstructure that uses a sequential approach, wherethe method first minimizes the energy usage and theminimum number of units using the Papoulias andGrossmann7 formulation and last uses the networksuperstructure to optimize the area cost. Floudas

and Ciric8 accounted for the area cost together withthe utility one in a simultaneous optimization proce-dure. These formulations involve inherent nonconvexi-ties that can lead to multiple local optimal solutions.Therefore, global search techniques under simpli-fied superstructures (i.e., isothermal mixers or nostream splits) were proposed to handle the HENSproblem.9-11

In the past decade, a set of papers accounted for theheat-exchanger design during the network synthesis.12-14

In the earlier paper, Polley et al.14 developed a relation-ship among the exchanger pressure drop, surface area,and heat-transfer coefficient, based on the well-knownDittus and Boelter correlation for the tube-side flow andon the Kern15 correlation for the shell-side flow. Theserelationships make possible a direct calculation of themain heat-exchanger parameters after setting the tubediameter, the number of tube passes, the tube pitch, thefluid allocation, and the tube arrangement. Thus, thetraditional successive simulation method for heat-exchanger design is avoided.

Nie and Zhu16 considered the pressure drop and heat-transfer enhancement in a retrofit design. The authorsused correlations that were similar to the ones byJegede and Polley12, to calculate the heat-transfercoefficients and the pressure drop for shell-and-tubeheat-exchanger units.

Liporace et al.17 studied the influence of the heat-exchanger design on the HENS using design correla-tions developed by Jegede and Polley12 in conjunc-tion with heuristics rules for network synthesis. Theyshowed that the level of detail used in the exchangerdesign can lead to different results in HENS problems.

* To whom correspondence should be addressed. Tel.: 1-412-2683642. Fax: 1-412-2687139. E-mail: [email protected].

† Fax: 55-21-25627425.

4019Ind. Eng. Chem. Res. 2003, 42, 4019-4027

10.1021/ie020965m CCC: $25.00 © 2003 American Chemical SocietyPublished on Web 07/17/2003

This result emphasizes that when developing HENSmethods, one must account for the detailed exchangerdesign.

Mizutani et al.18 presented an optimization model todesign shell-and-tube heat exchangers based on Bell-Delaware correlations to the shell-side fluid flow. Thismodel takes into account several detailed design vari-ables: number of tubes, number of tube passes, internaland external tube diameters, tube arrangement pattern,number of baffles, head type, and fluid allocation. Themodel is based on generalized disjunctive programmingand is optimized with a mixed-integer nonlinear pro-gramming (MINLP) reformulation to determinethe heat-exchanger design that minimizes the totalannualized cost accounting for area and pumping ex-penses.

The present work couples the heat-exchanger designmodel proposed by Mizutani et al.18 with the networksuperstructure given by Yee and Grossmann.9 Themodel is formulated with disjunctions for topologyselection, which in turn contains disjunctions for theheat-exchanger design, and solved using the logic-basedouter approximation.1

Problem Statement

Given are a set of hot and cold streams with theirsupply and target temperatures as well as their corre-sponding flow rates. Given are also hot and cold utilitytemperatures and their corresponding costs. For eachstream, the following physical properties are known:viscosity, density, thermal conductivity, and thermalcapacity. The problem then consists of determining theoptimal heat-exchanger network structure, the hot andcold utilities that are required, the heat loads of eachheat-exchanger unit and its design variables: numberof tubes, number of tube passes, internal and externaltube diameters, tube arrangement pattern, number ofbaffles, head type, and fluid allocation. The objective isto minimize the total annualized investment and oper-ating costs.

Problem Formulation

In the present work, the superstructure model pro-posed by Yee and Grossmann9 is considered. The modelassumes isothermal mixing in each stage of the super-structure, and as a result, all constraints become linearand consequently the problem turns well-behaved. Forsimplicity, we assume no stream splits as in ref 11.Therefore, the problem consists of optimizing the Yeeand Grossmann9 superstructure constraints togetherwith the heat-exchanger design constraints presentedby Mizutani et al.18 for each unit and minimizing thetotal annualized cost of area, utilities, and pumpingexpenses.

Consider a network superstructure with a candi-date set of heat-exchanger units whose existence is tobe determined through the Boolean variables Zijk torepresent the exchange of hot stream i with coldstream j in stage k. Also, consider that the detaileddesign of each selected unit is to be determined. The

following generalized representation form can be ap-plied:19

Note that the generalized form P1 contains three levelsof discrete decisions. The first level (Zijk) is on theexistence of units that defines the topology of thenetwork. The second level (Yd

design) is on the selection ofoptimal design decisions for the existing units thatcontain disjunctions (Yr

Re) to define design equationsrelated to the regime for the Reynolds number.

The above problem (P1) involves three types of vari-ables: x and c are continuous sets of variables, wherethe set x belongs to the network superstructure modelas temperatures and energy loads, among others, as wellas continuous variables from heat-exchanger designunits (i.e., number of tubes, shell diameters, etc.) and cis used to exclusively represent fixed charges; z and yare binary sets of variables, where the set z activatesor deactivates matches and controls the unit disjunc-tions and the set y is used in the design model; and Yand Z are Boolean variables related to the binaryvariables y and z, respectively.

The first set of constraints belongs to the Yee andGrossmann9 superstructure model, and these equationshold irrespective of the set of disjunctions that appliesfor heat-exchanger units. These constraints representthe logic relations for the Boolean variables Z in termsof the 0-1 variables z. The set of disjunctions M appliesfor the heat-exchanger units. If heat-exchanger unitexists (zijk ) 1), then the set of constraints that describesthat heat-exchanger design is enforced and a fixedcharge is applied; otherwise (zijk ) 0), the set of designconstraints does not apply and the fixed charge is setto zero. A detailed description of the heat-exchangerdesign model can be found in work by Mizutani et al.18

min f(x) + ∑ijk

cijk

s.t.

Asx + Bsz e as}Yee and Grossmann superstructure

[ Zijk

gijk(x) e 0

Adyd e ad

∨d∈D[Yd

design

gddesign(x) e 0

Aryr e ar

∨r∈R[Yr

Re0

grRe0(x) e 0 ] ]

cijk ) γijk

] ∨ [¬Zijkcijk ) 0 ] ijk ∈ M}

Hx design model

x ∈ X

y ∈ {0, 1}

z ∈ {0, 1}

Y ∈ {true, false}

Z ∈ {true, false} (P1)

4020 Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003

As shown in work by Mizutani et al.,18 the part of theproblem that deals with the detailed heat-exchangerdesign model can be reformulated as a MINLP problem.Therefore, the problem (P1) is then given by

which can be solved with the logic-based outer ap-proximation algorithm.1

Logic-Based Outer Approximation

The original outer approximation algorithm by Duranand Grossmann20 consists of solving nonlinear program-ming (NLP) subproblems (primal problem) that areobtained by fixing the binary variables and solvingmixed-integer linear programming (MILP) subproblems(master problem) that provide new values for the integervariables. For nonconvex problems, these subproblemsare solved until the primal problem does not improve.21

On the other hand, the logic-based outer approximationalgorithm1 reformulates the primal and master prob-lems in order to solve problems that rely on a logicrepresentation, in which mixed-integer logic is repre-sented through disjunctions and integer logic throughpropositions. One approach to solve problem P1 couldbe to formulate a primal problem that fixes all topology,z, and design, y, integer decisions and generate NLPsubproblems. This approach, however, tends to yieldinfeasible NLP subproblems.22 To avoid this difficulty,the proposed strategy consists of using primal MINLPproblems for fixed network topologies, but with heat-exchanger designs to be determined, and using masterproblems for the determination of new network topolo-gies as in the logic-based outer approximation algo-rithm.1

From the problem PHENS, the primal MINLP subprob-lem at iteration l, for a fixed chosen set of binary

variables z, is as follows:

It is important to point out that the primal problem Sl

relies on an MINLP problem because the integervariables y still remain from the detailed design model.Another important remark is that the subproblem Sl

requires only solutions of the design constraints thatbelong to those disjunctions in which their correspond-ing binary variable z indicates the existence of a matchunit (i.e., Zijk ) True). Therefore, the subproblem Sl

avoids the solution of the MINLP for the entire super-structure because a subset of the design equations forthe nonexisting units is removed from the model. Thishas the advantage of not only reducing the problemdimensionality but also of avoiding zero flows andenergy loads, which can lead to singularities.

The master problem is formulated with cumulativelinearizations of the nonlinear constraints that arederived from the optimal solutions of the primal sub-problems. The proposed master problem is transformedinto a MILP problem with the convex hull formulationof the disjunctions as follows:

The linearization set KLijk ) {l |Zijk

l ) True, l ) 1, ..., L}is defined for those linearizations generated in con-straints that are present in disjunctions in which theircorresponding Boolean variable Zijk is True.

The first set of constraint equations corresponds tothe cumulative linearizations of the objective functionthat is defined as the sum of the annualized area cost,

min f(x) + ∑ijk

cijk

s.t.

Asx + Bsz e as}Yee and Grossmann superstructure

[Zijk

hijk(x) e 0

Ady e ad

cijk ) γijk] ∨ [¬Zijk

cijk ) 0 ] ijk ∈ M}Hx design model

x ∈ X

y ∈ {0, 1}

z ∈ {0, 1}

Z ∈ {true, false} (PHENS)

min ZU ) f(x) + ∑ijk

cijk

s.t.

Asx + Bsz e 0

hijk(x) e 0Aijk

d x + Bijkd y e 0

cijk ) γijk} if Zijk

l ) True

cijk ) 0} if Zijkl ) False

x ∈ X

y ∈ {0, 1} (Sl)

min ZL ) Roa + ∑ijk

γijkzijk + ∑l)1,...,L

wlSl + ∑l∈K L

ijk

ijk∈M

wijkl Sijk

l

s.t.

f(xl) + ∇f(xl)T(x - xl) - Sl e Roa} l ) 1, ..., L

Tijkl {∇hijk(x

l)Tx + [hijk(xl) - ∇hijk(x

l)Txl]z} -

Sijkl e 0} l ∈ KL

ijk, ijk ∈ M

Asx + Bsz e 0

Aijkd x + Bijk

d y e 0

∑i∈ Bl

zi - ∑i∈ NBl

zi e |Bl| - 1} l ) 1, ..., L - 1

Sl, Sijkl g 0 (MHENS

l )

Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003 4021

the pumping cost, and the network utility cost:

where LMTD is the logarithm mean temperature dif-ference calculated by the Chen23 approximation.

To derive the nonlinear constraints for the optimalsolution from the primal subproblems, there are twopossibilities in the first term of eq 1. The first case iswhen the unit Zijk exists (i.e., Zijk ) True); then the heatload (Qijk), the overall heat-transfer coefficient (Uijk), andthe LMTD come from the optimal solution of the lastfeasible primal subproblem and are used to derive themaster problem cumulative constraints. The second caseis when the match unit Zijk does not exist (i.e., Zijk )False); in this case, the heat load is equal to zero, butthe overall heat-transfer coefficient and the LMTD arenot calculated in the primal subproblem. Consequently,there are two provisions in the implementation. Thefirst is to add a small constant to the heat load (e.g., ε

) 0.001) in order to avoid numerical problems in thederivatives; the second is to assume upper bound valuesto the overall heat-transfer coefficient and to the LMTD.In this way, the underestimation of the objective func-tion is guaranteed in the master problem formulation.

The second set of constraints in (MHENSl ) corresponds

to the cumulative linearizations of the design equationconstraints for each heat-exchanger unit activated inthe primal subproblems. Note that the convex-hullrepresentation is applied.

The third and the fourth sets of constraints cor-respond to the design equations and superstructureequations that are linear; they also contain binarydesign variables y and topology variables z, respectively.

Note that the master problem (MHENSl ) includes inte-

ger cut constraints in the topology binary variables inorder to exclude previous integer solutions. Also, be-cause the objective function and the design equationscontain nonconvexities, slack variables are introducedto avoid cutting off of the feasible integer solutions.21

Also, the stopping rule used is the heuristic terminationof no improvement in the primal subproblems.

Another important point in the model is the calcula-tion of the approach temperatures. In the originalformulation of Yee and Grossmann,9 the approachtemperatures were determined by the following con-straints:

The above formulation comes from the “big-M” repre-sentation, where Γ is a valid upper bound. The binaryvariables are used to activate or deactivate the con-straints for approach temperatures.

Note that these constraints can be expressed withonly “one-side” inequalities of the “big-M” representationbecause the cost of the exchangers decreases with highervalues for the temperature approaches dt. It can beverified by analyzing the Kuhn-Tucker conditions that

all the above constraints, in which zijk is equal to 1, areactive. The “other-side” inequalities would be as follows:

In the present work, the primal MINLP subproblemdoes not require the use of this “big-M” approachbecause all binary variables zijk are fixed. Therefore, thefollowing equality constraints apply:

On the other hand, in the master problem, the “big-M”formulation is used with the “other-side” inequalityconstraints (3); the reason is as follows. While the costof a heat-exchanger area decreases with higher valuesof the temperature approach, a reduction of such an areacan increase its pumping cost.

Proposed Algorithm

The proposed algorithm consists of solving succes-sively the MINLP primal subproblem (Sl), obtained froma given network topology and its design equations forthe assigned heat-exchanger units, and the MILPmaster subproblem (MHENS

l ), which is obtained fromthe cumulative linearizations of the objective functionand the design equations. The steps of the algorithmare as follows:

Step 1: Assume constant heat-transfer coefficientsand solve the original Yee and Grossmann9 superstruc-ture model. Set the result for the network topology asan initial guess and set k ) 0.

Step 2: Formulate the MINLP primal subproblemwith the initial network topology, including the designequations for the assigned units. Solve the problemusing a MINLP solver (i.e., DICOPT++). Set k ) k +1.

Step 3: Formulate the MILP master subproblemusing upper bounds for the heat-transfer coefficientsand LMTD that were not assigned in the last primalsubproblem.

Step 4: Formulate the MINLP primal subproblem,including the design equations for the assigned unitsin the last MILP master subproblem. Solve the problemusing an MINLP solver (i.e., DICOPT++). Set k ) k +1.

Step 5: If the objective of the MINLP primal sub-problem is higher, stop. Otherwise, go to step 3.

Examples

Three examples of increasing complexity are pre-sented to illustrate the proposed methodology of theHENS including heat-exchanger designs.

Example 1. The first example consists of determiningthe optimum heat-exchanger network for two hotstreams, two cold streams, one hot utility, and one coldutility in a two-stage superstructure. The detaileddesigns of the heat exchangers between process streamsare considered. However, for simplicity, a constantoverall heat-transfer coefficient is assumed betweenprocess and utility streams. Consequently, the logicprocedure turns off all design equations for the process-

f(x) ) ∑ijk

Kijk1 + Kijk

2 ( Qijk

UijkFtijkLMTDijk)κ

+ ∑ijk

Pijkcost +

Hcost∑j

Qhuj + Ccost∑i

Qcui (1)

dtijk e ti,k - tj,k + Γ(1 - zijk)dtijk+1 e ti,k+1 - tj,k+1 + Γ(1 - zijk) } ∀ ijk ∈ M (2)

-dtijk e -(ti,k - tj,k) + Γ(1 - zijk)-dtijk+1 e -(ti,k+1 - tj,k+1) + Γ(1 - zijk) }

∀ ijk ∈ M (3)

dtijk ) ti,k - tj,k

dtijk+1 ) ti,k+1 - tj,k+1 (4)


utility units and assumes the overall heat-transfercoefficient as a constant parameter for these units.

For simplicity, it is assumed that all unit designs haveone tube pass; therefore, it is not necessary to includecorrelations for the Ft temperature correction factor asdescribed by Mizutani et al.,18 and all temperaturecorrection factors are set to 1.

The data are shown in Table 1. Also, constant physicalproperties are considered for the process streams shownin Table 2.

The example was first solved by the original super-structure model (Yee and Grossmann9) in order toprovide an initial structure for the logic-based outerapproximation algorithm. This initial problem has 12discrete variables.

The results of the optimization problem provide theminimum cost configuration shown in Figure 1. Thecorresponding heat-exchanger detailed design data areshown in Table 3. As can be seen in Table 4, thisexample has 1089 equations, 1217 continuous variables,and 500 discrete variables, and the proposed logic-based

outer approximation algorithm finds the optimal solu-tion in three major iterations.

Table 5 shows how the heat-exchanger networkstructure as well as the heat-transfer coefficients changethrough the iterative solution of the proposed MINLPmodel. Our initial guess specifies the existence ofexchangers (1, 2) and (2, 1) with constant heat-transfercoefficients of 500 W/m2‚K and zero pumping cost. Thefirst primal problem then calculates a total cost of$96000/year, with most of it being due to utilities. Aftersolving the first primal problem, we calculate theoptimal heat exchanger for the specified tasks. Theresults show that the heat-transfer coefficient for ex-changer (1, 2) has increased by a factor of 4 and thatsome minor pumping costs have been added. Thisbehavior would also have been possible with a sequen-tial model that first identified the optimal networkstructure and then found the best individual heatexchangers. However, the simultaneous solution ap-proach presented allows the correction in heat-transfercoefficients to be fed back to the network structure.Thus, in the second and third primal problems, theactual heat-exchanger units selected are none of theones predicted by the first primal problem. The optimaldesign is found in iteration 2, with an annual cost of$95853/year, and in fact is the same topology as that inFigure 1 (note that exchanger (2, 1) was shifted to stage2).

Example 2. The second example consists of deter-mining the optimum heat-exchanger network for threehot streams, three cold streams, one hot utility, and onecold utility in a three-stage superstructure. Like inexample 1, a constant overall heat-transfer coefficientis assumed between process and utility streams. Thedata are shown in Table 6. Also, the same physicalproperties of example 1 were used.

The example was first solved by the original super-structure model (Yee and Grossmann9) in order toprovide an initial structure for the logic-based algo-rithm. This initial problem has 33 discrete variables.

The results of the optimization problem provide theminimum cost configuration shown in Figure 2. Thecorresponding detailed heat-exchanger design resultsare shown in Table 7. As can be seen in Table 8, thisexample has 3664 equations, 4534 continuous variables,and 1950 discrete variables, and the proposed logic-

Figure 1. Network structure for example 1.

Table 1. Data for Example 1a

m (kg/s) Tin (K) Tout (K)

H1 8.15 368 348H2 81.5 353 348C1 16.3 303 363C2 20.4 333 343CW 300 320S 500 500

a ∆Tmin ) 10 K. Area cost ) 1000 + 60A0.6$/year, where A )m2. Pumping cost ) 0.7(∆Ptmt/Ft + ∆Psms/Fs), where ∆P) Pa, m) kg/s, and F ) kg/m3. CW cost ) $6/kW‚year. S cost ) $60/kW‚year. Process-utilities overall heat-transfer coefficients ) 444W/m2‚K.

Table 2. Fluid Physical Properties for Example 1

viscosity, kg/m‚s 2.4 × 10-4

density, kg/m3 634thermal capacity, J/kg‚K 2454thermal conductivity, W/m‚K 0.114

Table 3. Results for Example 1

H1-C2 H2-C1

area (m2) 33.3 56.2U (W/m2‚K) 588 523no. of tubes 86 72shell diameter (m) 0.40 0.65internal/external tube diameter (mm) 21.18/25.40 46.58/50.80tube arrangement pattern square triangularno. of baffles 13 10head type fixed fixedhot fluid allocation shell tubes

Table 4. Summary of the Solver Results for Example 1

equations 1089continuous variables 1277discrete variables 500logic-based outer approximation iterations 3CPU time (s)a 14.9MINLP solver DICOPT++NLP solver CONOPT2MILP solver CPLEXa Pentium IV 1.5 GHz.


based outer approximation algorithm found the optimalsolution in five major iterations.

Example 3. The third example consists of determin-ing the optimum heat-exchanger network for seven hot

streams, three cold streams, one hot utility, and one coldutility in a six-stage superstructure. Like in example1, a constant overall heat-transfer coefficient is consid-ered between process and utility streams. The data areshown in Table 9. The same physical properties ofexample 1 were used.

The example was first solved by the original super-structure model (Yee and Grossmann9) in order toprovide an initial structure for the logic-based algo-rithm. This initial problem has 136 discrete vari-ables.

The results of the optimization problem provide theminimum cost configuration shown in Figure 3. Thecorresponding heat-exchanger detailed design data areshown in Tables 10 and 11. As can be seen in Table 12,this example has 16 939 equations, 20 408 continuousvariables, and 8452 discrete variables. The proposedlogic-based outer approximation algorithm found theoptimal solution in six major iterations.


Table 5. Iterative Solution Evolvement for Example 1

area (m2) U (W/m2‚K) area cost ($/year) pumping cost ($/year)

i, j stage 1 stage 2 stage 1 stage 2 stage 1 stage 2 stage 1 stage 2

Initial Guess (Constant Heat-Transfer Coefficients)1, 1 0.0 0.01, 2 0.0 38.1 500 15332, 1 57.2 0.0 500 16802, 2 0.0 0.0

First Primal Problem (Total Cost ) $96102/year, Utility Cost ) $90000/year)1, 1 0.0 0.01, 2 0.0 9.76 2008 1235 5852, 1 56.2 0.0 523 1673 1652, 2 0.0 0.0

Second Primal Problem (Total Cost ) $95853/year, Utility Cost ) $90000/year)1, 1 0.0 0.01, 2 0.0 33.3 587 1492 792, 1 0.0 56.2 523 1673 1652, 2 0.0 0.0

Third Primal Problem (Total Cost ) $129273/year, Utility Cost ) $123000/year)1, 1 0.0 13.1 611 1281 1931, 2 0.0 0.02, 1 0.0 0.02, 2 67.9 0.0 549 1754 145



H1 16.3 426 333H2 65.2 363 333H3 32.6 454 433C1 20.4 293 398C2 24.4 293 373C3 65.2 283 288CW 300 320S 700 700

a ∆Tmin ) 10 K. Area cost ) 1000 + 60A0.6$/year, where A )m2. Pumping cost ) 1.3(∆Ptmt/Ft + ∆Psms/Fs), where ∆P ) Pa, m) kg/s, and F ) kg/m3. CW cost ) $6/kW‚year. S cost ) $60/kW‚year. Process-utilities overall heat-transfer coefficients ) 444W/m2‚K.


Table 13 presents a comparison between the firstprimal and the final results with the proposed modelfor the three examples presented. In the first example,no heat matches changed between the first primalproblem and the final result and two heat matchesremained the same; therefore, as stated before, thetopology continued to be the same. One of the heatexchangers did not vary its heat-transfer coefficient andthe other decreased about 71%. In the second example,four heat matches changed and two remained the same;hence, the topology changed completely: one heat

exchanger increased its heat-transfer coefficient 4%, andthe other decreased about 19%. Finally, in the thirdexample, seven heat matches changed from the first


Table 7. Results for Example 2

u1 u2 u3 u4 u5 u6

area (m2) 141 88.8 13.9 47.8 8.8 8.5U (W/m2‚K) 658 859 514 544 889 582no. of tubes 581 228 24 80 11 11shell diameter (m) 0.58 0.61 0.41 0.59 0.40 0.39internal/external tube

diameter (mm)12.57/15.88 21.18/25.40 32.56/38.10 33.88/38.10 46.58/50.80 45.26/50.80

tube arrangement pattern square square square triangular square squareno. of baffles 11 8 7 7 7 7head type fixed fixed pull-through

floatingpull-through



floatinghot fluid allocation shell shell shell tubes shell tubes


equations 3664continuous variables 4534discrete variables 1950logic-based outer approximation iterations 5CPU time (s)a 171MINLP solver DICOPT++NLP solver CONOPT2MILP solver CPLEXa Pentium IV 1.5 GHz.



H1 134 413 313H2 235 433 393H3 12.1 483 318H4 28.5 533 333H5 102 553 483H6 14.2 623 443H7 38.9 653 433C1 235 543 658C2 143 403 543C3 104 293 403CW 293 298S 700 700

a ∆Tmin ) 10 K. Area cost ) 1000 + 60A0.6$/year, where A )m2. Pumping cost ) 0.7(∆Ptmt/Ft + ∆Psms/Fs), where ∆P) Pa, m) kg/s, and F ) kg/m3. CW cost ) $6/kW‚year. S cost ) $60/kW‚year. Process-utilities overall heat-transfer coefficients ) 444W/m2‚K.


primal problem to the final result; only one match didnot vary, and its heat-transfer coefficient decreased 18%.

Conclusions

In this paper, an optimization model has been pro-posed for the synthesis of heat-exchanger networksincluding shell-and-tube heat-exchanger designs. Themain contribution of this work is the inclusion of adetailed design model in the optimization of the heat-exchanger network superstructure in order to simulta-neously determine the network topology and the designof its heat-exchanger units.

The model is based on generalized disjunctive pro-gramming and is optimized with the logic-based outerapproximation method. Upper bounds for the overallheat-transfer coefficients and LMTD were included for

the nonselected matches in order to underestimate theobjective function.

The application and usefulness of the proposed methodhave been shown in three example problems of increas-ing complexity. The results indicate that the methodol-ogy can properly account for the tradeoffs between area,pumping, and utility costs during the HENS includingshell-and-tube heat-exchanger designs.

Acknowledgment

The authors acknowledge financial support providedby CNPq (National Council of Science and TechnologicalDevelopment) and NSF Equipment Grant CTS-0094407.

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(6) Floudas, C. A.; Ciric, A. R.; Grossmann, I. E. AutomaticSynthesis of Optimum Heat Exchanger Network Configuration.AIChE J. 1986, 32 (2), 276.

(7) Papoulias, S. A.; Grossmann, I. E. A Structural Optimiza-tion Approach in Process Synthesis. Part II: Heat Recovery.

(8) Floudas, C. A.; Ciric, A. R. Strategies for OvercomingUncertainties in Heat Exchanger Network Synthesis. Comput.Chem. Eng. 1989, 13 (10), 1133-1152.

(9) Yee, T. F.; Grossmann, I. E. Simultaneous OptimizationModels for Heat Integration. II. Heat Exchanger Network Syn-thesis. Comput. Chem. Eng. 1990, 14 (10), 1165-1184.

(10) Quesada, I.; Grossmann, I. E. Global Optimization Algo-rithm for Heat Exchanger Networks. Ind. Eng. Chem. Res. 1993,32, 487-499.

(11) Zamora, J. M.; Grossmann, I. E. A Global MINLP Opti-mization Algorithm for the Synthesis of Heat Exchanger Networkswith no Stream Splits. Comput. Chem. Eng. 1998, 22 (3), 367-384.

Table 10. Results for Example 3, Units 1-4

u1 u2 u3 u4

area (m2) 57.8 618 78.2 330U (W/m2‚K) 786 602 516 568no. of tubes 74 2,116 100 1,130shell diameter (m) 0.81 1.18 0.84 0.93internal/external tube diameter (mm) 45.26/50.80 14.83/19.05 46.58/50.80 14.83/19.05tube arrangement pattern square triangular square triangularno. of baffles 7 8 20 7head type pull-through floating fixed fixed split-ring floatinghot fluid allocation shell shell shell shell

Table 11. Results for Example 3, Units 5-8

u5 u6 u7 u8

area (m2) 49.5 219 179 178U (W/m2‚K) 545 427 594 700no. of tubes 63 561 460 228shell diameter (m) 0.76 0.91 0.84 1.21internal/external tube diameter (mm) 45.26/50.80 21.18/25.40 21.18/25.40 45.26/50.80tube arrangement pattern square square square squareno. of baffles 7 11 8 8head type pull-through floating fixed fixed fixedhot fluid allocation shell shell shell shell


equations 16939continuous variables 20408discrete variables 8452logic-based outer approximation iterations 6CPU time (s)a 1175MINLP solver DICOPT++NLP solver CONOPT2MILP solver CPLEXa Pentium IV 1.5 GHz.

Table 13. Comparison between the First Primal and theFinal Results

examplechanges in thesuperstructure

overall heat-transfercoefficient changesfor the unchanged

matches

1 no heat match changed -71% and 0%two heat matches did

not changeone heat match shifted in the

superstructure stages2 four heat matches changed +4% and -19%

two heat matches did not changeone heat match shifted in the

superstructure stages3 seven heat matches changed -18%

one heat match did not changeone heat match shifted in the

superstructure stages


(12) Jegede, F. O.; Polley, G. T. Optimum Heat ExchangerDesign. Trans. Inst. Chem. Eng. 1992, 70 (A2), 133-141.

(13) Polley, G. T.; Panjeh Shahi, M. H. M. Interfacing HeatExchanger Network Synthesis and Detailed Heat ExchangerDesign. Trans. Inst. Chem. Eng. 1991, 69, 445-457.

(14) Polley, G. T.; Panjeh Shahi, M. H. P.; Jegede, F. O.Pressure Drop Considerations in the Retrofit of Heat ExchangerNetworks. Trans. Inst. Chem. Eng. 1990, 68, 211-220.

(15) Kern, D. Q. Process Heat Transfer; McGraw-Hill: NewYork, 1950.

(16) Nie, X. R.; Zhu, X. X. Heat Exchanger Network RetrofitConsidering Pressure Drop and Heat-Transfer Enhancement.AIChE J. 1999, 45 (6), 1239-1254.

(17) Liporace, F. S.; Pessoa, F. L. P.; Queiroz, E. M. TheInfluence of Heat Exchanger Design on the Synthesis of HeatExchanger Networks. Braz. J. Chem. Eng. 2000, 17 (4-7), 735-750.

(18) Mizutani, F. T.; Pessoa, F. L. P.; Queiroz, E. M.; Hauan,S.; Grossmann, I. E. Mathematical Programming Model for theOptimal Design of Shell and Tube Heat Exchangers. Submittedfor publication.

(19) Raman, R.; Grossmann, I. E. Modelling and ComputationalTechniques for Logic Based Integer Programming. Comput. Chem.Eng. 1994, 7, 563-578.

(20) Duran, M. A.; Grossmann, I. E. An Outer ApproximationAlgorithm for a Class of Mixed Integer Nonlinear Programs. Math.Prog. 1994, 66, 327-350.

(21) Viswanathan, J.; Grossmann, I. E. A Combined PenaltyFunction and Outer-Approximation method for MINLP Optimiza-tion. Comput. Chem. Eng. 1990, 14, 769-782.

(22) Kocis, G. R.; Grossmann, I. E. A Modeling and Decomposi-tion Strategy for the MINLP Optimization of Process Flowsheets.Comput. Chem. Eng. 1989, 13, 797-819.

(23) Chen, J. J. Letter to the Editor: Comments on improve-ment on a replacement for the logarithmic mean. Chem. Eng. Sci.1987, 42, 2488.

Received for review December 3, 2002Revised manuscript received May 27, 2003

Accepted May 29, 2003

IE020965M


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