Mathematical Programming Model for Heat-Exchanger Network

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

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Mathematical Programming Model for Heat-Exchanger Network Synthesis Including Detailed Heat-Exchanger Designs. 2. Network Synthesis Fabio T. Mizutani,† Fernando L. P. Pessoa,† and Eduardo M. Queiroz† Departamento de Engenharia Quı´mica, Escola de Quı´mica, 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 includes a heat-exchanger design model. This model takes into account several detailed design variables: number of tubes, number of tube passes, internal and external tube diameters, tube arrangement pattern, number of baffles, head type, and fluid allocation (i.e., to the shell or tubes). The network superstructure with individual heat-exchanger designs is solved using the logic-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 for topology selection, which in turn has disjunctions for the heat-exchanger design. The proposed model determines the heat-exchanger network that minimizes the total annualized cost accounting for area, pumping, and utility expenses. Examples are presented to illustrate this method. Introduction Heat-exchanger network synthesis (HENS) has been the subject of a significant research over the last 40 years.2 Several papers in the chemical process synthesis area have developed mathematical programming tools to solve this problem. However, most formulations assume constant heat-transfer coefficients and counterflow arrangement for all stream matches, which can lead to nonoptimal results because they usually are far from the operational reality because heat-transfer coefficients are strongly influenced by the exchanger geometry and multipass shell-and-tube units are frequently found in chemical process plants. Gundersen and Naess3 presented an extensive HENS review, and recently Furman and Sahinidis2 reported that over 400 papers have been published on the subject over the last 4 decades. The minimum utility target was properly defined in the literature because Linnhoff and Hindmarsh4 proposed the energy cascade algorithm in the well-known pinch-point method. At the same time, several mathematical programming methods have been proposed to solve the problem. Papoulias and Grossmann5,7 tackled the minimum energy usage and the minimum number of units. Floudas et al.6 formulated a superstructure that uses a sequential approach, where the method first minimizes the energy usage and the minimum number of units using the Papoulias and Grossmann7 formulation and last uses the network superstructure to optimize the area cost. Floudas * To whom correspondence should be addressed. Tel.: 1-4122683642. Fax: 1-412-2687139. E-mail: [email protected]. † Fax: 55-21-25627425.

and Ciric8 accounted for the area cost together with the utility one in a simultaneous optimization procedure. These formulations involve inherent nonconvexities that can lead to multiple local optimal solutions. Therefore, global search techniques under simplified superstructures (i.e., isothermal mixers or no stream splits) were proposed to handle the HENS problem.9-11 In the past decade, a set of papers accounted for the heat-exchanger design during the network synthesis.12-14 In the earlier paper, Polley et al.14 developed a relationship among the exchanger pressure drop, surface area, and heat-transfer coefficient, based on the well-known Dittus and Boelter correlation for the tube-side flow and on the Kern15 correlation for the shell-side flow. These relationships make possible a direct calculation of the main heat-exchanger parameters after setting the tube diameter, the number of tube passes, the tube pitch, the fluid allocation, and the tube arrangement. Thus, the traditional successive simulation method for heatexchanger design is avoided. Nie and Zhu16 considered the pressure drop and heattransfer enhancement in a retrofit design. The authors used correlations that were similar to the ones by Jegede and Polley12, to calculate the heat-transfer coefficients and the pressure drop for shell-and-tube heat-exchanger units. Liporace et al.17 studied the influence of the heatexchanger design on the HENS using design correlations developed by Jegede and Polley12 in conjunction with heuristics rules for network synthesis. They showed that the level of detail used in the exchanger design can lead to different results in HENS problems.

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

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

This result emphasizes that when developing HENS methods, one must account for the detailed exchanger design. Mizutani et al.18 presented an optimization model to design shell-and-tube heat exchangers based on BellDelaware correlations to the shell-side fluid flow. This model takes into account several detailed design variables: number of tubes, number of tube passes, internal and external tube diameters, tube arrangement pattern, number of baffles, head type, and fluid allocation. The model is based on generalized disjunctive programming and is optimized with a mixed-integer nonlinear programming (MINLP) reformulation to determine the heat-exchanger design that minimizes the total annualized cost accounting for area and pumping expenses. The present work couples the heat-exchanger design model proposed by Mizutani et al.18 with the network superstructure given by Yee and Grossmann.9 The model is formulated with disjunctions for topology selection, which in turn contains disjunctions for the heat-exchanger design, and solved using the logic-based outer approximation.1

following generalized representation form can be applied:19

min f(x) +

cijk ∑ ijk

[ [ ]] } s.t.

Asx + Bsz e as}

Yee and Grossmann superstructure

Zijk gijk(x) e 0

Adyd e ad Ydesign d gdesign (x) e 0 d

∨ Aryr e ar d∈D 0 YRe r ∨ Re0 r∈R gr (x) e 0 cijk ) γijk

[

]

[

¬Z ∨ c ijk ijk ) 0

]

ijk ∈ M

Hx design model

x∈X Problem Statement Given are a set of hot and cold streams with their supply and target temperatures as well as their corresponding flow rates. Given are also hot and cold utility temperatures and their corresponding costs. For each stream, the following physical properties are known: viscosity, density, thermal conductivity, and thermal capacity. The problem then consists of determining the optimal heat-exchanger network structure, the hot and cold utilities that are required, the heat loads of each heat-exchanger unit and its design variables: number of tubes, number of tube passes, internal and external tube diameters, tube arrangement pattern, number of baffles, head type, and fluid allocation. The objective is to minimize the total annualized investment and operating costs.

Problem Formulation In the present work, the superstructure model proposed by Yee and Grossmann9 is considered. The model assumes isothermal mixing in each stage of the superstructure, and as a result, all constraints become linear and consequently the problem turns well-behaved. For simplicity, we assume no stream splits as in ref 11. Therefore, the problem consists of optimizing the Yee and Grossmann9 superstructure constraints together with the heat-exchanger design constraints presented by Mizutani et al.18 for each unit and minimizing the total annualized cost of area, utilities, and pumping expenses. Consider a network superstructure with a candidate set of heat-exchanger units whose existence is to be determined through the Boolean variables Zijk to represent the exchange of hot stream i with cold stream j in stage k. Also, consider that the detailed design of each selected unit is to be determined. The

y ∈ {0, 1} z ∈ {0, 1} Y ∈ {true, false} Z ∈ {true, false}

(P1)

Note that the generalized form P1 contains three levels of discrete decisions. The first level (Zijk) is on the existence of units that defines the topology of the ) is on the selection of network. The second level (Ydesign d optimal design decisions for the existing units that contain disjunctions (YRe r ) to define design equations related to the regime for the Reynolds number. The above problem (P1) involves three types of variables: x and c are continuous sets of variables, where the set x belongs to the network superstructure model as temperatures and energy loads, among others, as well as continuous variables from heat-exchanger design units (i.e., number of tubes, shell diameters, etc.) and c is used to exclusively represent fixed charges; z and y are binary sets of variables, where the set z activates or deactivates matches and controls the unit disjunctions and the set y is used in the design model; and Y and Z are Boolean variables related to the binary variables y and z, respectively. The first set of constraints belongs to the Yee and Grossmann9 superstructure model, and these equations hold irrespective of the set of disjunctions that applies for heat-exchanger units. These constraints represent the logic relations for the Boolean variables Z in terms of the 0-1 variables z. The set of disjunctions M applies for the heat-exchanger units. If heat-exchanger unit exists (zijk ) 1), then the set of constraints that describes that heat-exchanger design is enforced and a fixed charge is applied; otherwise (zijk ) 0), the set of design constraints does not apply and the fixed charge is set to zero. A detailed description of the heat-exchanger design model can be found in work by Mizutani et al.18

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

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

min f(x) +

variables z, is as follows:

min ZU ) f(x) + s.t.

cijk ∑ ijk

Asx + Bsz e 0 hijk(x) e 0 Adijkx + Bdijky e 0 cijk ) γijk

s.t. Ax+Bzea} s

s

s

Yee and Grossmann superstructure

[ ][ Zijk hijk(x) e 0 Ady e ad cijk ) γijk

¬Z ∨ c ijk ijk ) 0

]

cijk ∑ ijk

}

cijk ) 0}

if Zlijk ) True

if Zlijk ) False x∈X

ijk ∈ M

y ∈ {0, 1}

Hx design model

x∈X y ∈ {0, 1} z ∈ {0, 1} Z ∈ {true, false}

}

(PHENS)

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

Logic-Based Outer Approximation The original outer approximation algorithm by Duran and Grossmann20 consists of solving nonlinear programming (NLP) subproblems (primal problem) that are obtained by fixing the binary variables and solving mixed-integer linear programming (MILP) subproblems (master problem) that provide new values for the integer variables. For nonconvex problems, these subproblems are solved until the primal problem does not improve.21 On the other hand, the logic-based outer approximation algorithm1 reformulates the primal and master problems in order to solve problems that rely on a logic representation, in which mixed-integer logic is represented through disjunctions and integer logic through propositions. One approach to solve problem P1 could be to formulate a primal problem that fixes all topology, z, and design, y, integer decisions and generate NLP subproblems. This approach, however, tends to yield infeasible NLP subproblems.22 To avoid this difficulty, the proposed strategy consists of using primal MINLP problems for fixed network topologies, but with heatexchanger designs to be determined, and using master problems for the determination of new network topologies as in the logic-based outer approximation algorithm.1 From the problem PHENS, the primal MINLP subproblem at iteration l, for a fixed chosen set of binary

(Sl)

It is important to point out that the primal problem Sl relies on an MINLP problem because the integer variables y still remain from the detailed design model. Another important remark is that the subproblem Sl requires only solutions of the design constraints that belong to those disjunctions in which their corresponding binary variable z indicates the existence of a match unit (i.e., Zijk ) True). Therefore, the subproblem Sl avoids the solution of the MINLP for the entire superstructure because a subset of the design equations for the nonexisting units is removed from the model. This has the advantage of not only reducing the problem dimensionality but also of avoiding zero flows and energy loads, which can lead to singularities. The master problem is formulated with cumulative linearizations of the nonlinear constraints that are derived from the optimal solutions of the primal subproblems. The proposed master problem is transformed into a MILP problem with the convex hull formulation of the disjunctions as follows:

min ZL ) Roa +

γijkzijk + ∑ wlSl + ∑ wlijkSlijk ∑ ijk l)1,...,L ijk

l∈KL ijk ∈ M

s.t. f(xl) + ∇f(xl)T(x - xl) - Sl e Roa}

l ) 1, ..., L

Tlijk{∇hijk(xl)Tx + [hijk(xl) - ∇hijk(xl)Txl]z} Slijk e 0}

l ∈ Kijk L , ijk ∈ M

Asx + Bsz e 0 Adijkx + Bdijky e 0

∑ zi - ∑ zi e |Bl| - 1}

i∈ Bl

l ) 1, ..., L - 1

i∈ NBl

Sl, Slijk g 0

(MlHENS)

l The linearization set Kijk L ) {l | Zijk ) True, l ) 1, ..., L} is defined for those linearizations generated in constraints that are present in disjunctions in which their corresponding Boolean variable Zijk is True. The first set of constraint equations corresponds to the cumulative linearizations of the objective function that is defined as the sum of the annualized area cost,

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the pumping cost, and the network utility cost:

f(x) )

∑ ijk

K1ijk + K2ijk

(

Qijk

UijkFtijkLMTDijk H

cost

)

κ

+

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

Pcost ∑ ijk + ijk

∑j Qhuj + C ∑i Qcui cost

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

where LMTD is the logarithm mean temperature difference calculated by the Chen23 approximation. To derive the nonlinear constraints for the optimal solution from the primal subproblems, there are two possibilities in the first term of eq 1. The first case is when the unit Zijk exists (i.e., Zijk ) True); then the heat load (Qijk), the overall heat-transfer coefficient (Uijk), and the LMTD come from the optimal solution of the last feasible primal subproblem and are used to derive the master problem cumulative constraints. The second case is when the match unit Zijk does not exist (i.e., Zijk ) False); in this case, the heat load is equal to zero, but the overall heat-transfer coefficient and the LMTD are not calculated in the primal subproblem. Consequently, there are two provisions in the implementation. The first is to add a small constant to the heat load (e.g., ) 0.001) in order to avoid numerical problems in the derivatives; the second is to assume upper bound values to the overall heat-transfer coefficient and to the LMTD. In this way, the underestimation of the objective function is guaranteed in the master problem formulation. l The second set of constraints in (MHENS ) corresponds to the cumulative linearizations of the design equation constraints for each heat-exchanger unit activated in the primal subproblems. Note that the convex-hull representation is applied. The third and the fourth sets of constraints correspond to the design equations and superstructure equations that are linear; they also contain binary design variables y and topology variables z, respectively. l ) includes inteNote that the master problem (MHENS ger cut constraints in the topology binary variables in order to exclude previous integer solutions. Also, because the objective function and the design equations contain nonconvexities, slack variables are introduced to avoid cutting off of the feasible integer solutions.21 Also, the stopping rule used is the heuristic termination of no improvement in the primal subproblems. Another important point in the model is the calculation of the approach temperatures. In the original formulation of Yee and Grossmann,9 the approach temperatures were determined by the following constraints:

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

}

}

∀ ijk ∈ M (3)

In the present work, the primal MINLP subproblem does not require the use of this “big-M” approach because all binary variables zijk are fixed. Therefore, the following equality constraints apply:

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

(4)

On the other hand, in the master problem, the “big-M” formulation is used with the “other-side” inequality constraints (3); the reason is as follows. While the cost of a heat-exchanger area decreases with higher values of the temperature approach, a reduction of such an area can increase its pumping cost. Proposed Algorithm The proposed algorithm consists of solving successively the MINLP primal subproblem (Sl), obtained from a given network topology and its design equations for the assigned heat-exchanger units, and the MILP l master subproblem (MHENS ), which is obtained from the cumulative linearizations of the objective function and the design equations. The steps of the algorithm are as follows: Step 1: Assume constant heat-transfer coefficients and solve the original Yee and Grossmann9 superstructure model. Set the result for the network topology as an initial guess and set k ) 0. Step 2: Formulate the MINLP primal subproblem with the initial network topology, including the design equations for the assigned units. Solve the problem using a MINLP solver (i.e., DICOPT++). Set k ) k + 1. Step 3: Formulate the MILP master subproblem using upper bounds for the heat-transfer coefficients and LMTD that were not assigned in the last primal subproblem. Step 4: Formulate the MINLP primal subproblem, including the design equations for the assigned units in the last MILP master subproblem. Solve the problem using an MINLP solver (i.e., DICOPT++). Set k ) k + 1. Step 5: If the objective of the MINLP primal subproblem is higher, stop. Otherwise, go to step 3. Examples

∀ ijk ∈ M (2)

The above formulation comes from the “big-M” representation, where Γ is a valid upper bound. The binary variables are used to activate or deactivate the constraints for approach temperatures. Note that these constraints can be expressed with only “one-side” inequalities of the “big-M” representation because the cost of the exchangers decreases with higher values for the temperature approaches dt. It can be verified by analyzing the Kuhn-Tucker conditions that

Three examples of increasing complexity are presented to illustrate the proposed methodology of the HENS including heat-exchanger designs. Example 1. The first example consists of determining the optimum heat-exchanger network for two hot streams, two cold streams, one hot utility, and one cold utility in a two-stage superstructure. The detailed designs of the heat exchangers between process streams are considered. However, for simplicity, a constant overall heat-transfer coefficient is assumed between process and utility streams. Consequently, the logic procedure turns off all design equations for the process-

Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003 4023 Table 3. Results for Example 1 (m2)

area U (W/m2‚K) no. of tubes shell diameter (m) internal/external tube diameter (mm) tube arrangement pattern no. of baffles head type hot fluid allocation

H1-C2

H2-C1

33.3 588 86 0.40 21.18/25.40 square 13 fixed shell

56.2 523 72 0.65 46.58/50.80 triangular 10 fixed tubes

Table 4. Summary of the Solver Results for Example 1 equations continuous variables discrete variables logic-based outer approximation iterations CPU time (s)a MINLP solver NLP solver MILP solver a

Figure 1. Network structure for example 1. Table 1. Data for Example 1a H1 H2 C1 C2 CW S

m (kg/s)

Tin (K)

Tout (K)

8.15 81.5 16.3 20.4

368 353 303 333 300 500

348 348 363 343 320 500

a ∆T 0.6 min ) 10 K. Area cost ) 1000 + 60A $/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 ) 444 W/m2‚K.

Table 2. Fluid Physical Properties for Example 1 viscosity, kg/m‚s density, kg/m3 thermal capacity, J/kg‚K thermal conductivity, W/m‚K

2.4 × 10-4 634 2454 0.114

utility units and assumes the overall heat-transfer coefficient as a constant parameter for these units. For simplicity, it is assumed that all unit designs have one tube pass; therefore, it is not necessary to include correlations for the Ft temperature correction factor as described by Mizutani et al.,18 and all temperature correction factors are set to 1. The data are shown in Table 1. Also, constant physical properties are considered for the process streams shown in Table 2. The example was first solved by the original superstructure model (Yee and Grossmann9) in order to provide an initial structure for the logic-based outer approximation algorithm. This initial problem has 12 discrete variables. The results of the optimization problem provide the minimum cost configuration shown in Figure 1. The corresponding heat-exchanger detailed design data are shown in Table 3. As can be seen in Table 4, this example has 1089 equations, 1217 continuous variables, and 500 discrete variables, and the proposed logic-based

1089 1277 500 3 14.9 DICOPT++ CONOPT2 CPLEX

Pentium IV 1.5 GHz.

outer approximation algorithm finds the optimal solution in three major iterations. Table 5 shows how the heat-exchanger network structure as well as the heat-transfer coefficients change through the iterative solution of the proposed MINLP model. Our initial guess specifies the existence of exchangers (1, 2) and (2, 1) with constant heat-transfer coefficients of 500 W/m2‚K and zero pumping cost. The first primal problem then calculates a total cost of $96000/year, with most of it being due to utilities. After solving the first primal problem, we calculate the optimal heat exchanger for the specified tasks. The results show that the heat-transfer coefficient for exchanger (1, 2) has increased by a factor of 4 and that some minor pumping costs have been added. This behavior would also have been possible with a sequential model that first identified the optimal network structure and then found the best individual heat exchangers. However, the simultaneous solution approach presented allows the correction in heat-transfer coefficients to be fed back to the network structure. Thus, in the second and third primal problems, the actual heat-exchanger units selected are none of the ones predicted by the first primal problem. The optimal design is found in iteration 2, with an annual cost of $95853/year, and in fact is the same topology as that in Figure 1 (note that exchanger (2, 1) was shifted to stage 2). Example 2. The second example consists of determining the optimum heat-exchanger network for three hot streams, three cold streams, one hot utility, and one cold utility in a three-stage superstructure. Like in example 1, a constant overall heat-transfer coefficient is assumed between process and utility streams. The data are shown in Table 6. Also, the same physical properties of example 1 were used. The example was first solved by the original superstructure model (Yee and Grossmann9) in order to provide an initial structure for the logic-based algorithm. This initial problem has 33 discrete variables. The results of the optimization problem provide the minimum cost configuration shown in Figure 2. The corresponding detailed heat-exchanger design results are shown in Table 7. As can be seen in Table 8, this example has 3664 equations, 4534 continuous variables, and 1950 discrete variables, and the proposed logic-

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Figure 2. Network structure for example 2. Table 5. Iterative Solution Evolvement for Example 1 U (W/m2‚K)

area (m2) i, j

stage 1

stage 2

stage 1

area cost ($/year)

stage 2

stage 1

stage 2

pumping cost ($/year) stage 1

stage 2

Initial Guess (Constant Heat-Transfer Coefficients) 1, 1 1, 2 2, 1 2, 2

0.0 0.0 57.2 0.0

0.0 38.1 0.0 0.0

1, 1 1, 2 2, 1 2, 2

0.0 0.0 56.2 0.0

First Primal Problem (Total Cost ) $96102/year, Utility Cost ) $90000/year) 0.0 9.76 2008 1235 0.0 523 1673 165 0.0

1, 1 1, 2 2, 1 2, 2

0.0 0.0 0.0 0.0

1, 1 1, 2 2, 1 2, 2

0.0 0.0 0.0 67.9

500 500

Second Primal Problem (Total Cost ) $95853/year, Utility Cost ) $90000/year) 0.0 33.3 587 1492 56.2 523 1673 0.0 Third Primal Problem (Total Cost ) $129273/year, Utility Cost ) $123000/year) 13.1 611 1281 0.0 0.0 0.0 549 1754 145

Table 6. Data for Example 2a H1 H2 H3 C1 C2 C3 CW S

1533 1680

m (kg/s)

Tin (K)

Tout (K)

16.3 65.2 32.6 20.4 24.4 65.2

426 363 454 293 293 283 300 700

333 333 433 398 373 288 320 700

a ∆T 0.6 min ) 10 K. Area cost ) 1000 + 60A $/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 ) 444 W/m2‚K.

based outer approximation algorithm found the optimal solution in five major iterations. Example 3. The third example consists of determining the optimum heat-exchanger network for seven hot

585

79 165

193

streams, three cold streams, one hot utility, and one cold utility in a six-stage superstructure. Like in example 1, a constant overall heat-transfer coefficient is considered between process and utility streams. The data are shown in Table 9. The same physical properties of example 1 were used. The example was first solved by the original superstructure model (Yee and Grossmann9) in order to provide an initial structure for the logic-based algorithm. This initial problem has 136 discrete variables. The results of the optimization problem provide the minimum cost configuration shown in Figure 3. The corresponding heat-exchanger detailed design data are shown in Tables 10 and 11. As can be seen in Table 12, this example has 16 939 equations, 20 408 continuous variables, and 8452 discrete variables. The proposed logic-based outer approximation algorithm found the optimal solution in six major iterations.

Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003 4025

Figure 3. Network structure for example 3. Table 7. Results for Example 2 (m2)

u1

u2

u3

u4

u5

u6

area U (W/m2‚K) no. of tubes shell diameter (m) internal/external tube diameter (mm) tube arrangement pattern no. of baffles head type

141 658 581 0.58 12.57/15.88

88.8 859 228 0.61 21.18/25.40

13.9 514 24 0.41 32.56/38.10

47.8 544 80 0.59 33.88/38.10

8.8 889 11 0.40 46.58/50.80

8.5 582 11 0.39 45.26/50.80

square 11 fixed

square 8 fixed

hot fluid allocation

shell

shell

square 7 pull-through floating shell

triangular 7 pull-through floating tubes

square 7 pull-through floating shell

square 7 pull-through floating tubes


3664 4534 1950 5 171 DICOPT++ CONOPT2 CPLEX

Pentium IV 1.5 GHz.

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

Table 9. Data for Example 3a H1 H2 H3 H4 H5 H6 H7 C1 C2 C3 CW S

m (kg/s)

Tin (K)

Tout (K)

134 235 12.1 28.5 102 14.2 38.9 235 143 104

413 433 483 533 553 623 653 543 403 293 293 700

313 393 318 333 483 443 433 658 543 403 298 700

a ∆T 0.6 min ) 10 K. Area cost ) 1000 + 60A $/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 ) 444 W/m2‚K.

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

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Table 10. Results for Example 3, Units 1-4 (m2)


u1

u2

u3

u4

57.8 786 74 0.81 45.26/50.80 square 7 pull-through floating shell

618 602 2,116 1.18 14.83/19.05 triangular 8 fixed shell

78.2 516 100 0.84 46.58/50.80 square 20 fixed shell

330 568 1,130 0.93 14.83/19.05 triangular 7 split-ring floating shell

Table 11. Results for Example 3, Units 5-8 (m2)


u5

u6

u7

u8

49.5 545 63 0.76 45.26/50.80 square 7 pull-through floating shell

219 427 561 0.91 21.18/25.40 square 11 fixed shell




16939 20408 8452 6 1175 DICOPT++ CONOPT2 CPLEX

Pentium IV 1.5 GHz.

Acknowledgment

Table 13. Comparison between the First Primal and the Final Results

changes in the superstructure

example 1

2

3

no heat match changed two heat matches did not change one heat match shifted in the superstructure stages four heat matches changed two heat matches did not change one heat match shifted in the superstructure stages seven heat matches changed one heat match did not change one heat match shifted in the superstructure stages

the nonselected matches in order to underestimate the objective function. The application and usefulness of the proposed method have been shown in three example problems of increasing complexity. The results indicate that the methodology can properly account for the tradeoffs between area, pumping, and utility costs during the HENS including shell-and-tube heat-exchanger designs.

overall heat-transfer coefficient changes for the unchanged matches -71% and 0%

+4% and -19%

-18%

primal problem to the final result; only one match did not vary, and its heat-transfer coefficient decreased 18%. Conclusions In this paper, an optimization model has been proposed for the synthesis of heat-exchanger networks including shell-and-tube heat-exchanger designs. The main contribution of this work is the inclusion of a detailed design model in the optimization of the heatexchanger network superstructure in order to simultaneously determine the network topology and the design of its heat-exchanger units. The model is based on generalized disjunctive programming and is optimized with the logic-based outer approximation method. Upper bounds for the overall heat-transfer coefficients and LMTD were included for

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Received for review December 3, 2002 Revised manuscript received May 27, 2003 Accepted May 29, 2003 IE020965M

Mathematical Programming Model for Heat-Exchanger Network

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