Synthesis of Heat Exchanger Networks with Optimal Placement of

Ind. Eng. Chem. Res. 2010, 49, 2849–2856

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Synthesis of Heat Exchanger Networks with Optimal Placement of Multiple Utilities Jose´ M. Ponce-Ortega,† Medardo Serna-Gonza´lez,† and Arturo Jimeńez-Gutie´rrez*,‡ Facultad de Ingenierıá Quı´mica, UniVersidad Michoacana de San Nicola´s de Hidalgo, Morelia, Michoacań 58060, Me´xico, and Departamento de Ingenierıá Quı´mica, Instituto Tecnolo´gico de Celaya, Celaya, Gto. 38010, Me´xico

The synthesis problem of optimal heat exchanger networks using multiple utilities is addressed. A mixed integer nonlinear programming (MINLP) model is developed based on a stagewise superstructure that contains all possible matches between hot and cold streams in every stage. Unlike previous MINLP formulations, exchanges of process streams with utilities are allowed in each stage of the superstructure to determine the optimal location of hot and cold utilities. The optimal choice of different types of available utilities is considered through a disjunctive programming formulation. The model can handle forbidden matches and required and restricted matches as well as isothermal and nonisothermal process streams. The MINLP problem is readily solved using commercial local solvers. Four examples are presented to show the application of the proposed methodology. No computation complications were observed in the examples presented, and they required small CPU times. 1. Introduction The optimal synthesis of heat exchanger networks (HENs) has been widely studied during the past 40 years because of their economic and environmental benefits. Gundersen and Naess,1 Jezowski,2,3 and Furman and Sahinidis4 have reported comprehensive reviews of methodologies to obtain optimal HEN. In general, the most common approaches to solve the HEN synthesis problem can be classified as sequential and simultaneous methods. Sequential synthesis methods based on pinch analysis5-7 and on the use of mathematical programming8-11 have been developed. Simultaneous approaches formulate the HEN synthesis problem as a mixed integer nonlinear programming (MINLP) problem.12,13 Most of the current methods for heat exchanger networks assume that only one type of hot utility and one type of cold utility are available. Also, utility exchangers are typically placed at the extremes of the network, since utilities are assumed to be hotter or colder than the hottest or coldest process stream. Many times, however, several hot and cold utilities are available. For example, furnace flue gas, steam at different levels (high, medium, and low pressure), hot oil circuit, and hot water can be employed as hot utilities, while refrigeration at different levels, air, and water can be used as cold utilities. Under these conditions, the total annual cost of a HEN depends on the heat loads and types of utilities used. Pinch technology has been used for targeting optimal loads and levels for multiple utilities, based on the process grand composite curve,14 on an extension of the problem table algorithm,15 or on a direct numerical geometric-based technique.16 These procedures determine the consumption target of each utility by maximizing the use of the cheapest utilities and minimizing the loads of expensive ones. Papoulias and Grossmann11 formulated a transshipment model to calculate the minimum utility cost for multiple utilities at constant temperature (i.e., point utilities). A method for solving the minimum utility * To whom correspondence should be addressed. Tel.: (+52-461) 611-7575. Fax: (+52-461) 611-7744. E-mail: [email protected]. † Universidad Michoacana de San Nicola´s de Hidalgo. ‡ Instituto Tecnolo´gico de Celaya.

cost problem for multiple utilities at constant and not constant temperature was developed by Viswanathan and Evans17 based on the use of the out-of-kilter algorithm. Jezowski and Friedler18 proposed basic algorithms to calculate the minimum utility cost for problems with multiple utilities with forbidden matches without the use of mathematical programming techniques. Shethna et al.19 proposed an MILP transportation formulation based on temperature intervals to predict trade-offs between capital and utility operating costs for networks with multiple utilities and forbidden matches. The load on each utility was taken as an optimization variable. However, they used linear approximations for the cost of heat exchangers and for logmean temperature differences in order to obtain a simplified linear objective function. A targeting procedure was proposed by Shenoy et al.20 to determine the cost-optimal loads for multiple utilities. This methodology is based on the pinch analysis and depends on the cheapest utility principle. Once the optimal targets are known, the network is synthesized using the pinch design method for tasks with multiple pinches. The initial design may be evolved to reduce the number of units and optimized to arrive at the final network. A limitation of this methodology is that it cannot consider forbidden stream matches or disallowed heat flows. Recently, Isafiade and Fraser21 formulated the synthesis problem of HENs as an MINLP problem, which can handle match restrictions, multiple utilities, and utilities available over a range of temperatures. This formulation used an interval-based MINLP superstructure (IBMS) similar to the stagewise representation of Yee and Grossmann.13 Although the IBMS method can simultaneously trade-off energy, heat transfer area and number of units, it has three limitations. First, since the interval boundaries of the superstructure are set by the terminal temperatures of either the hot or the cold set of streams, superstructure intervals are not uniquely defined. Second, by assuming the temperature interval boundaries as fixed data, this approach eliminates potential network configurations that may contain the optimal one. Third, since utilities are regarded as process streams to define the superstructure intervals, the heat capacity flow rates of utilities are set as optimization variables,

10.1021/ie901750a  2010 American Chemical Society Published on Web 02/05/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 6, 2010

Figure 1. Superstructure for the HEN synthesis for two hot and two cold process streams.

so that heat balances for utilities show bilinear terms that may prevent a global optimum solution to be obtained. In this work, a simultaneous method for synthesizing HENs with multiple utilities is presented. The method consists of a stagewise superstructure that allows intermediate placement of the utilities in the network. The types and heat loads of the available utilities are treated as optimization variables. Unlike previous methods for the synthesis of HENs with multiple utilities, the model can handle isothermal process streams by extending the disjunctive programming formulation recently reported for problems with single utilities by Ponce-Ortega et al.22 2. Outline of the Proposed Model Figure 1 shows a schematic representation of the superstructure proposed for two hot and two cold process streams. This superstructure is similar to the stagewise representation by Yee and Grossmann,13 in which the problem is divided into stages; each stage includes the possibility to exchange heat between each pair of hot and cold process streams, and the border temperatures for each stage are treated as optimization variables. To avoid nonlinear heat balances, the model assumes isothermal mixing at the exit of each stage of the superstructure. Notice in the representation of Figure 1 that utilities are allowed to be placed within each stage of the superstructure (situation not allowed in the original Yee and Grossmann13 representation). The intermediate location of the utilities in the superstructure can yield important savings when different types of utilities are available. The problem addressed in this paper is stated as follows: Given a set of hot and cold process streams with their supply and target temperatures, as wells as their flow rates, heat capacities, and film heat transfer coefficients, and a set of hot and cold utilities with different temperature levels, film heat transfer coefficients and unit costs, the problem consists in determining the optimal HEN structure and the type of utilities used to yield the HEN with the minimum total annual cost.

3. Model Formulation The model formulation consists of a set of constraints used to model the superstructure shown in Figure 1. The following sets are used in the model formulation. HPS represents the hot process streams, CPS represents the cold process streams, and ST is the number of stages in the superstructure. HPS1 and HPS2 are subsets for the hot process streams for the nonisothermal and isothermal streams, respectively (HPS ) HPS1∪HPS2), whereas CPS1 and CPS2 are subsets for the nonisothermal and isothermal cold process streams (CPS ) CPS1∪CPS2). HU and CU are the sets for hot and cold utilities. The indexes i, j, and k are used for the hot process streams, cold process streams, and a stage in the superstructure, respectively; m and n are indexes for cold and hot utilities. All symbols used are defined in the nomenclature section, and the model is described as follows. a. Overall Energy Balance for Each Process Stream. Hot process streams exchange their overall heat loads with cold process streams and/or with cold utilities in each stage of the superstructure, (TIN,i - TOUT,i)FCpi )

∑ ∑

qi,j,k +

k∈ST j∈CPS

Fλcond ) i

∑ ∑

qi,j,k +

k∈ST j∈CPS

∑ ∑

k∈ST m∈CU

qcui,km,

i ∈ HPS1

∑ ∑

k∈ST m∈CU

qcui,km,

(1)

i ∈ HPS2

(2) Similarly, energy integration and hot utilities can be used in each stage of the superstructure to treat the cold process streams, (TOUT,j - TIN,j)FCpj )

∑ ∑

qi,j,k +

k∈ST i∈HPS

Fλevap ) j

∑ ∑

k∈ST i∈HPS

qi,j,k +

∑ ∑q

k∈ST n∈HU

j ∈ CPS1

∑ ∑q

k∈ST n∈HU

n , huj,k

n , huj,k

(3)

j ∈ CPS2

(4) and Fλevap are the total latent heats In these equations, Fλcond i j for condensation and evaporation for streams i and j.


b. Energy Balance for Each Stage. Energy balances are required to calculate the temperatures in each border of the superstructure for the process streams. For the hot process streams, the heat exchanged in each stage is transferred to cold process streams and/or cold utilities, (ti,k - ti,k+1)FCpi )

∑

∑

qi,j,k +

j∈CPS

m∈CU

qcui,km,

k ∈ ST, i ∈ HPS1

∑

∑q

qi,j,k +

i∈HPS

n∈CU

n , huj,k

i ∈ HPS

TOUT,i ) ti,NOK+1,

i ∈ HPS

TIN,j ) tj,NOK+1, TOUT,j ) tj,1,

j ∈ CPS j ∈ CPS

dti,j,k+1 e ti,k+1 - tj,k+1 + ∆Tmax i ∈ HPS, i,j (1 - zi,j,k), j ∈ CPS, k ∈ ST

(19)

i ∈ HPS, k ∈ ST

max dtcui,k+1 e ti,k+1 - TIN,cui,k + ∆Tcu (1 - zcui,k), i

(20)

i ∈ HPS, k ∈ ST (21)

j ∈ CPS,

k ∈ ST

(22) (6)

(7)

max dthuj,k+1 e TOUT,huj,k - tj,k+1 + ∆Thu (1 - zhuj,k), j

j ∈ CPS, k ∈ ST (23)

where ∆Tmax is an upper limit for the temperature difference. These equations are activated when the heat exchanger exists; when the heat exchanger does not exist, the parameter ∆Tmax is used to satisfy the constraints. A proper estimation of the upper limit for temperature differences, ∆Tmax, is carried out through the following condition,

(8)

if TIIN,i - TJIN,j < ∆Tmin

(9)

∆Tmax i,j ) abs[TIIN,i - TJIN,j] + ∆Tmin else ∆Tmax i,j ) max{0, TIIN,i - TJIN,j, TJOUT,j - TIOUT,i}

(10)

d. Temperature Feasibility. To get a monotonous decrease of temperature from the left side to the right side of the superstructure, the following constraints are included,

(24)

In addition, a minimum approach temperature is included to avoid infinite areas (or extremely large values), ∆Tmin g dti,j,k,

i ∈ HPS,

j ∈ CPS,

k ∈ ST

(25)

ti,k g ti,k+1,

k ∈ ST,

i ∈ HPS1

(11)

∆Tmin g dtcui,k,

i ∈ HPS,

k ∈ ST

(26)

ti,k ) TIN,i,

k ∈ ST,

i ∈ HPS2

(12)

∆Tmin g dthuj,k,

j ∈ CPS,

k ∈ ST

(27)

tj,k g tj,k+1,

k ∈ ST,

j ∈ CPS1

(13)

tj,k ) TIN,j,

k ∈ ST,

j ∈ CPS2

(14)

e. Definition for the Heat Exchangers. To determine the existence of heat exchangers, the following set of equations are needed: qi,j,k - Qmax i,j zi,j,k e 0, qcui,km -

(18)

max dthui,k e TIN,huj,k - tj,k + ∆Thu (1 - zhui,k), j

Notice that these types of equations are not needed for the isothermal process streams because their temperatures do not change along the superstructure. c. Assignment of Inlet Temperatures to the Superstructure. The inlet and outlet temperatures of the superstructure are specified from the stream data as follows, TIN,i ) ti,1,

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i ∈ HPS, j ∈ CPS, k ∈ ST

(5)

k ∈ ST, j ∈ CPS1

- zi,j,k),

max dtcui,k e ti,k - TOUT,cui,k + ∆Tcu (1 - zcui,k), i

whereas, for the cold process streams, their thermal processing is achieved with hot process streams and/or hot utilities, (tj,k - tj,k+1)FCpj )

dti,j,k e ti,k - tj,k +

∆Tmax i,j (1

i ∈ HPS,

j ∈ CPS,

k ∈ ST (15)

e 0,

i ∈ HPS,

k ∈ ST,

m ∈ CU (16)

qhuj,kn - Qmax n e 0, j zhuj,k

j ∈ CPS,

k ∈ ST,

n ∈ HU (17)

Qmax m i zcui,k

Here, the binary variables z, zcu, and zhu are activated when there is a heat load higher than zero, and Qmax corresponds to max is set as the the upper limit for heat transfer. The value of Qi,j smallest heat content of the two streams involved in the match, with the values of Qmax and Qmax corresponding to the total heat i j load for the streams i and j. f. Temperature Differences. The temperatures differences are calculated as follows,

g. Selection of the Type of Cold Utility. For different types of cold utilities, the optimization approach must select the optimal one according to the following disjunction,

[[

][

Zcui,k Zcui,k1 Zcui,k2 TOUT,cui,k ) TOUT, cu1 ∨ TOUT,cui,k ) TOUT, cu2 TIN,cui,k ) TIN, cu1 TIN,cui,k ) TIN, cu2

[

Zcui,kNCU ∨ TOUT,cui,k ) TOUT, cuNCU TIN,cui,k ) TIN, cuNCU

]] [

]

∨ ···

]

¬Zcui,k ∨ TOUT,cui,k ) 0 , TIN,cui,k ) 0 i ∈ HPS,

k ∈ ST

Here, when the Boolean variable Zcui,k is true, a cooler is required for hot stream i in stage k, then one of the Boolean variables Zcuikm must be true to select the proper type of cold utility along with its inlet and outlet temperatures. On the other hand, when the cooler is not required, Zcui,k is false and the temperatures of the cold utility are simply set as zero. Notice that TOUT,cui,k and TIN,cui,k are optimization variables whereas TOUT,cum, and TIN,cum, are fixed parameters for each cold utility available, which are known previous to the optimization process.

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The disjunction is modeled using the convex hull reformulation23 as follows: zcui,k ) TOUT,cui,k ) TIN,cui,k )

∑

m∈CU

min

zcui,km,

∑

m∈CU

∑

m∈CU

i ∈ HPS,

tout,cui,km,

i ∈ HPS,

i ∈ HPS,

tin, cui,km ) TIN, cui,km zcui,km,

k ∈ ST

i ∈ HPS,

tin,cui,km,

tout, cui,km ) TOUT, cum zcui,km,

i ∈ HPS,

(28)

k ∈ ST

(29)

k ∈ ST

k ∈ ST, k ∈ ST,

(30)

m ∈ CU (31) m ∈ CU (32)

Notice that these equations are linear. h. Selection of the Type of Hot Utility. The following disjunction is used for the selection of the type of hot utility,

[[

][

Zhuj,k Zhuj,k1 Zhuj,k2 TOUT,huj,k ) TOUT, hu1 ∨ TOUT,huj,k ) TOUT, hu2 TIN,huj,k ) TIN, hu1 TIN,huj,k ) TIN, hu2

[

Zhuj,kNHU ∨ · · · ∨ TOUT,huj,k ) TOUT, huNHU TIN,huj,k ) TIN, huNHU

][

]

]

¬Zhuj,k ∨ TOUT,huj,k ) 0 , TIN,huj,k ) 0 k ∈ ST

The Boolean variables Zhuj,k and Zhuj,kn perform the same role as the ones used for the selection of the cold utility. Here, TOUT,huj,k and TIN,huj,k are optimization variables, whereas TOUT,hun and TIN,hun are fixed parameters for each hot utility available. Using the convex hull reformulation, one obtains the following set of equations:

TOUT,huj,k ) TIN,huj,k )

∑z

n∈HU

n , huj,k

∑t

n , out,huj,k

n∈HU

∑t

n∈HU

n , in,huj,k

tout, huj,kn ) TOUT, hun zhuj,kn, tin, huj,kn ) TIN, huj,kn zhuj,kn,

j ∈ CPS,

k ∈ ST

j ∈ HPS, j ∈ CPS,

j ∈ CPS, j ∈ CPS,

(33)

k ∈ ST k ∈ ST

k ∈ ST, k ∈ ST,

∑ ∑ ∑C

(34)

(35)

n ∈ HU (36) n ∈ CU

I. Objective Function. The objective function is formulated as the minimization of the total annual cost, which includes the operating costs for hot and cold utilities, the fixed costs for exchangers, coolers, and heaters, and the capital cost due to heat exchanger area for each heat transfer unit. The objective function considers the optimal selection of the type of utility for each match when the utility is needed; therefore, the unit cost and film heat transfer coefficient for each type of utility is included. Chen’s approximation24 is used for the calculation of the long mean temperature differences. The objective function

∑ ∑ ∑C

+

m CUmqcui,k

i∈HPS k∈ST m∈CU

n HUnqhuj,k

+

j∈CPS k∈ST n∈HU

.

∑ ∑ ∑C

Fi,jz,j,k

i∈HPS j∈CPS k∈ST

∑ ∑ ∑C ∑ ∑C ∑ ∑

+

∑ ∑C

Chu,j

Fi,cuzcui,k

i∈HPS k∈ST

.

+

∑ ∑C

Fcu,jzhuj,k

j∈CPS k∈ST

( (

)

+

}

β 1 1 + hi,k hj,k + 1/3 dti,j,k + dti,j,k+1 +δ (dti,j,k)(dti,j,k+1) 2 . β 1 1 qcui,km + m h hcu i m∈CU + 1/3 dtcui,k + dtcui,k+1 (dtcui,k)(dtcui,k+1) +δ 2 . β 1 1 qhuj,kn n + h hhu j n∈HU 1/3 dthuj,k + dthuj,k+1 (dthuj,k)(dthuj,k+1) +δ 2

qi,j,k

{[ {[

i,cu

i∈HPS k∈CU

{[

i,j

i∈HPS j∈CPS k∈ST

j∈CPS k∈HU

j ∈ CPS,

zhuj,k )

can be written as follows,

∑

(

∑

(

(

) ]

)

) ] ) ) ]

(

} }

(37)

where CCUm and CHUn are the unit costs for the different cold and hot utilities. In summary, the optimization model consists in minimizing eq 37 subject to constraints given by eqs 1-36. Remarks. (i) The model includes the optimal selection for different types of hot and cold utilities. (ii) A disjunctive formulation is developed for the optimal selection of utilities. (iii) All the constraints in the model are linear; the objective function includes a nonlinear function for the calculation of the capital cost of the exchangers. (iv) The model incorporates both isothermal and nonisothermal process streams. 4. Examples Four problems are presented to show the applicability of the proposed method. For all examples, the solver DICOPT in addition with CPLEX and CONOPT included in the general algebraic modeling system (GAMS) software25 were used. Example 1. This is a fairly simple problem used as a motivational example. The problem consists of two hot and two cold process streams, with the operating data given in Table 1. Only one hot and one cold type of utilities are available, with unit costs of 80 $/kW-year and 15 $/kW-year, respectively. The capital cost for the heat exchangers are determined by Cexc ) 5500 + 150A, where A is in m2 and Cexc is in $/year. The minimum temperature difference allowed for the matches is 20 K. The problem was solved first using the superstructure of Yee and Grossmann,13 with utilities placed at the extremes of the superstructure. Figure 2 shows the solution thus obtained. The model consists of 61 constraints, with 12 binary variables. The problem was solved in 1.11 s of CPU time in a Pentium Table 1. Data for Example 1 stream

TIN [K]

TOUT [K]

FCp [kW K-1]

h [kW/(m2 K)]

H1 H2 HU C1 C2 CU

423 363 410 293 298 293

333 333 410 398 373 323

20 80

0.1 0.2 2.5 0.2 0.3 1.0

25 30


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Figure 2. Solution for Example 1 with Yee and Grossmann model. Figure 4. Optimal network configuration for Example 2. Table 3. Results Comparison for Example 2 solution method

this paper

Shenoy et al.20

Isafiade and Fraser21

HPS load [kW] MPS load [kW] LPS load [kW] CW load [kW] number of units TAC [$/year]

238.7 0 151.3 740.0 7 97,079

203 53 119.5 725.5 9 98,263

244.6 1 143.7 739.3 9 97,211

Table 4. Data for Example 3 cost stream TIN [°C] TOUT [°C] FCp [kW K-1] h [kW/(m2 K)] ($/(kW year) Figure 3. Optimal solution for Example 1. Table 2. Data for Example 2 cost stream TIN [°C] TOUT [°C] FCp [kW K-1] h [kW/(m2 K)] ($/(kW year) H1 H2 HPSa MPSb LPSc C1 CWd a

105 185 210 160 130 25 5

25 35 209 159 129 185 6

High pressure steam. steam. d Cooling water.

10 5

7.5 b

0.5 0.5 5.0 5.0 5.0 0.5 2.6

Medium pressure steam.

160 110 50 10 c

Low pressure

IV at 2.8 GHz. The solution to this problem includes three heat exchangers between process streams, two coolers, and two heaters to yield utilities and capital costs of $149,000/year and $247,316/year, for a total cost of $396,316/year. When the proposed model was used, the solution shown in Figure 3 was obtained. Notice that the optimal placement of the hot utility used to heat the cold process streams C1 is located in the middle of the network, between two process-process heat exchangers. The utility and investment costs for this configuration are $101,500/year and $283,846/year, to yield a total annual cost of $385,346/year. It can be seen that in this case the optimal placement for the utilities yields savings of 2.8% respect to solution obtained with the Yee and Grossmann13 model. Example 2. This example was reported by Shenoy et al.20 and also solved by Isafiade and Fraser.21 The problem consists of two hot and one cold process streams, with high, medium and low pressure steam available as hot utilities, and cold water as a cold utility. The streams data are shown in Table 2. To determine the capital cost for the heat exchanger units, the cost model Cexc ) 800[Area (m2)] was used, with an annualization factor of 0.298/year. Figure 4 was obtained with the application of the proposed algorithm. The network requires three heat exchangers for energy integration. There are also two coolers and two heaters, one that uses 238.7 kW of high pressure steam and the other one 151.3 kW of the low pressure steam. Notice that there is a

H1 H2 HPSa MPSb LPSc C1 C2 C3 CWd ACe

155 230 255 205 150 115 50 60 30 40

85 40 254 204 149 210 180 175 40 65

150 85

140 55 60

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

a High pressure steam. b Medium pressure steam. steam. d Cooling water. e Air cooling.

70 50 20

10 5 c

Low pressure

heater in stage three of the superstructure, a situation not allowed in the original model by Yee and Grosmmann.13 The network reported in Figure 4 has a total investment cost of $43,922/ year and a total utility cost of $53,157/year to yield a total annual cost of $97.079/year. Table 3 shows a result comparison between the solution obtained here with respect to the solutions reported by Shenoy et al.20 and Isafiade and Fraser21 for the same example. Notice that the solution obtained in this work uses the highest amount of high pressure steam but has the lowest number of units, and it shows savings in total annual cost of 1.22% and 0.14% with respect to the solutions reported by Shenoy et al.20 and Isafiade and Fraser.21 It is worth noting that the optimal solution was readily obtained with the proposed formulation. Example 3. This example was also taken from the works by Shenoy et al.20 and Isafiade and Fraser.21 The problem consists of two hot and three cold process streams, three different types of steam (high, medium, and low pressure steam) as hot utilities, and cooling water and cooling air as cold utilities. The stream data are reported in Table 4. Figure 5 shows the network obtained with the proposed approach. Three heat exchanger units between process streams are needed. For cooling utilities, two coolers that use cold water are required for the hot process streams H1 and H2. For hot utilities, a heater for the cold process stream C1 that uses high pressure steam and two heaters for the cold process streams C2 and C3 that use medium pressure steam are selected. The total heat transfer area for the eight units of the network reported in Figure 5 is 4926.7 m2, to yield a total capital cost of $540,459/

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Figure 5. Optimal network configuration for Example 3. Table 5. Results Comparison for Example 3 solution method

this paper

Shenoy et al.20

Isafiade and Fraser21

HPS load [kW] MPS load [kW] LPS load [kW] CW load [kW] AC load [kW] number of units TAC [$/year]

4290.0 4075.3 0 7665.3 0 8 1,121,175

4885 3575 0 3600 4160 9 1,158,500

5928.5 1852.0 0 7080.5 0 7 1,150,460 Figure 6. Optimal network for Example 4.

Table 6. Data for Example 4 cost stream TIN [K] TOUT [K] Fλ [kW K-1] h [kW/(m2 K)] ($(kW year) H1 H2 H3 H4 HPSa MPSb LPSc C1 C2 C3 CWd ACe

340 390 420 475 627 473 423 350 375 400 303 313

340 390 420 475 627 473 423 350 375 400 315 338

1900.0 1493.1 2594.4 1999.1

992.5 1801.2 4361.6

problem

1.52 1.63 1.75 1.58 2.5 2.0 1.5 1.81 1.72 1.64 1.0 0.5

a High pressure steam. b Medium pressure steam. steam. d Cooling water. e Air cooling.

Table 7. Problem Size and Computing Time for Each Case Study

100 50 20

10 5 c

Low pressure

year. The total utility cost is $580,716/year, and the total annual cost for the network is $1,121,175/year. Table 5 shows a comparison of results with the ones reported by Shenoy et al.20 and Isafiade and Fraser.21 It can be seen that the network of Figure 5 is the one with the lowest consumption for the high pressure steam and with the highest consumption of cooling water. The total annual cost of the network obtained with the proposed method is 3.3% and 2.6% cheaper with respect to the solutions reported by Shenoy et al.20 and Isafiade and Fraser,21 respectively. Example 4. In this example, the use of the model to handle isothermal process streams with multiple utilities is considered. The isothermal process stream data reported by Ponce-Ortega et al.22 was taken as a basis (only one type of hot and cold utility was considered in such work). The data for the problem is reported in Table 6. The capital cost for the heat exchanger units is calculated as Cexc ) 1650A0.65, with A in m2 and an annualization factor of 0.23 year-1. Figure 6 shows the solution obtained with the proposed approach. In terms of the use of

Example Example Example Example

1 2 3 4

constraints

binary variables

continuous variables

CPU time [sec]

61 511 1391 789

12 120 360 216

53 501 1442 861

1.11 1.24 43.67 0.88

utilities, cooling water was needed to treat hot process stream H1, while low pressure steam was selected for the cold process stream C3. The network requires seven units with a total heat transfer area of 374 m2, which yields an investment cost of $34,417/year. With a utility cost of $40,374/year, the network shows a total annual cost of $74,791/year. Table 7 shows the size of each problem as well as the CPU time required to solve them in a Pentium IV at 2.8 GHz. One can notice that the computing time was relatively small. 4. Conclusions An MINLP model for the optimal HEN synthesis involving different types of utilities considering simultaneously the capital cost for heat exchangers and the operating cost for utilities has been presented. The formulation determines the optimal location of the utility units within the network, as opposed to their placement at the extremes of the superstructure as typically done in reported methodologies. The optimal selection for the type of utilities was modeled through a disjunctive formulation, and then reformulated into an MINLP problem using the convex hull technique. Overall, the model consists of a set of linear constraints with a nonlinear objective function; this structure aids the solution process, such that optimal solutions are readily obtained using local optimization solvers such as DICOPT. In addition, the solution to the case studies shows that the proposed formulation can provide better results than those reported with other methodologies.


Nomenclature A ) heat transfer area C ) area cost coefficient CCUm ) unit cost of cold utility CHUn ) unit cost of hot utility CF ) fixed charge for exchangers Cp ) specific heat capacity CPS ) {j | j is a cold process stream} CPS1 ) {j | j is a non isothermal cold process stream} CPS2 ) {j | j is an isothermal cold process stream} CU ) cold utility dti,j,k ) temperature approach difference for match (i,j) at temperature location k dtcui,k ) temperature approach difference for match between hot stream i and a cold utility at the temperature location k dthuj,k ) temperature approach difference for match between cold stream j and a hot utility at the temperature location k F ) flow rate FCp ) heat capacity flow rate h ) fouling heat transfer coefficient HPS ) {i | i is a hot process stream} HPS1 ) {i | i is a non isothermal hot process stream} HPS2 ) {i | i is an isothermal hot process stream} HU ) hot utility NOK ) total number of stages qi,j,k ) heat exchanged between hot process stream i and cold process stream j in stage k m ) heat exchanged between cold utility m and hot stream i qcui,k

in stage k n ) heat exchanged between hot utility n and cold stream j qhuj,k

in stage k Qmax ) upper bound for heat exchange ST ) {k | k is a stage in the superstructure, k ) 1, ..., NOK} ti,k ) temperature of hot stream i at the hot end of stage k tj,k ) temperature of cold stream j at the hot end of stage k TINi ) inlet temperature of stream i TOUTi ) outlet temperature of stream i TOUTcui,k ) outlet temperature for cold utility in stage k TINcuj,k ) inlet temperature for cold utility in stage k TOUT,cum ) parameter for the outlet temperature for cold utility m TIN,cum ) parameter for the inlet temperature for cold utility m m ) disaggregated variables for the outlet temperature for tout,cui,k

cold utility m m ) disaggregated variables for the inlet temperature for tin,cui,k

cold utility m TOUThuj,k ) outlet temperature for hot utility in stage k TIN,huj,k ) inlet temperature for hot utility in stage k TOUT,hun ) parameter for the outlet temperature for hot utility n TIN,hun ) parameter for the inlet temperature for hot utility n n ) disaggregated variables for the outlet temperature for tout,huj,k

hot utility n n ) disaggregated variables for the inlet temperature for tin,huj,k

hot utility n ∆Tmax ) upper bound for temperature difference ∆TMIN ) minimum approach temperature difference Z ) boolean variables used to model disjunctions zi,j,k ) binary variables for match (i,j) in stage k m ) binary variables for match between cold utility m and zcui,k

hot stream i

zcunj,k ) binary variables for match between hot utility n and cold

stream j Greek Symbols Fλicond ) condensation heat load for hot stream i Fλjevap ) evaporation heat load for cold stream j

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β ) exponent for area in cost equation δ ) small number Subscripts and Superscripts i ) hot process stream j ) cold process stream k ) index for stage (1, ..., NOK) and temperature location (1, ..., NOK + 1) m ) cold utility n ) hot utility

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ReceiVed for reView November 4, 2009 ReVised manuscript receiVed January 8, 2010 Accepted January 20, 2010 IE901750A

Synthesis of Heat Exchanger Networks with Optimal Placement of

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