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Ind. Eng. Chem. Res. 2004, 43, 6766-6773
Two-Level Optimization Algorithm for Heat Exchanger Networks Including Pressure Drop Considerations Medardo Serna-Gonza´ lez,† Jose´ Marı´a Ponce-Ortega,† and Arturo Jime´ nez-Gutie´ rrez*,‡ Facultad de Ingenierı´a Quı´mica, Universidad Michoacana de San Nicola´ s de Hidalgo, Avenida Francisco J. Mu´ jica S/N, Ciudad Universitaria, Edificio M, Morelia, Mich. 58060 Me´ xico, and Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´ gico de Celaya, Avenida Tecnolo´ gico y Garcı´a Cubas S/N, Celaya, Gto. 38010 Me´ xico
A mixed-integer nonlinear programming (MINLP) algorithm to account for the effects of the allowable pressure drops for process streams on the structure and cost of heat exchanger networks is presented. Because MINLP models for large problems typically show significant convergence problems, the synthesis model is decomposed into two levels. For a given energy recovery level, an MINLP model is formulated in an inner loop using assumed values for the film heat-transfer coefficients for each stream. In the outer loop, with the use of the areas provided by the MINLP model and thermal-hydraulic models that relate the exchanger area, pressure drops, and film coefficients, updated values of the film coefficients are calculated for each stream according to the specified values for allowable pressure drops. The algorithm provides a network that makes use of the allowable pressure drops specified for the process streams at the total minimum yearly cost. The algorithm shows excellent convergence properties, can incorporate restrictions on matches between streams, and is appropriate for problems with significant differences in the values of film coefficients. Three case studies are used to show the application of the proposed method. 1. Introduction The synthesis of heat exchanger networks has been a major problem in the area of process design, promoted heavily during the 1970s because of the rising energy costs observed during that decade. Two types of methods stand as the main approaches for this problem. One is the use of the pinch-point concept, through which a heat exchanger network (HEN) with minimum energy consumption for a given value of ∆Tmin, the minimum approach temperature difference, can be obtained.1 The second involves the use of superstructures, which typically require a mixed-integer nonlinear programming (MINLP) model for their solution, as shown, for example, by Yee and Grossmann.2 The formulation for the synthesis of HENs has mainly been done considering two major aspects: the energy consumed in the form of utilities and the capital investment required for the heat exchangers. In principle, the best combination of these two factors can be identified by any of the above-mentioned approaches. Until recently, however, the contribution of pressure drop aspects had been neglected. Because allowable pressure drops are commonly specified in the design stage of heat exchanger units, the incorporation of such a detail in the design of HENs provides an approach of practical importance. Polley et al.3 were among the first to point out the relevance of including pressure drop considerations in industrial projects involving HENs. Polley and Panjeh Shahi4 * To whom correspondence should be addressed. Tel.: (+52461) 611-7575 ext. 139. Fax: (+52-461) 611-7744. E-mail:
[email protected]. † Universidad Michoacana de San Nicola´s de Hidalgo. ‡ Instituto Tecnolo´gico de Celaya.
reported a method that included pressure drops in the definition of a heat exchanger network problem. The method was based on the combination of a minimumarea algorithm (based on composite curve diagrams) and algebraic equations that relate the heat exchanger area to the pressure drops and film heat-transfer coefficients. Through this formulation, a yearly cost for the system that includes the cost of utilities, the capital investment in heat exchangers, and the cost of pumping equipment can be obtained. Although the pinch-point method provides a convenient tool for the design and analysis of HENs, two shortcomings of the formulation based on pinch concepts can be mentioned. First, the area target procedure5 is valid only for a set of hot and cold streams with the same values of film heat-transfer coefficients, as it is based on vertical heat transfer between the composite curves. Second, it does not allow for thermal coupling restrictions between two streams. The first point provides a limitation when significant differences in heattransfer coefficients are observed (for example, when the process involves both liquid and gas streams), and the second would be important when two streams are required not to exchange heat or when values for the exchanger area or heat duty for a given exchange are specified. In this work, an algorithm for the synthesis of heat exchanger networks is presented. The method is based on an MINLP formulation that includes the allowable pressure drops for the streams instead of using fixed values for the film heat-transfer coefficients. The original MINLP model is likely to show convergence problems, particularly for moderate- to large-size problems. To overcome this difficulty, the global problem is decomposed into two nested levels. In the inner loop,
10.1021/ie0497700 CCC: $27.50 © 2004 American Chemical Society Published on Web 09/18/2004
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dimensionless factor fi is equal to 1.0 when stream i flows through the shell side; otherwise, it is equal to Dt/Dti. To simplify the model, the clean film coefficients (hci or hcj) can be eliminated by substituting eq 2 into eq 1. For any stream i, the pressure drop relationship can then be written as
∆Pi ) KiAci Figure 1. Contact area for a heat exchanger.
an MINLP model such as the one suggested by Yee and Grossmann2 is used, taking as a basis assumed values of the film heat-transfer coefficients for the streams. As a result, an optimum HEN structure is obtained for a given heat recovery value, along with the contact areas for the streams. In the outer loop, the method uses mathematical expressions that relate the heat-transfer areas to pressure drops and film heat-transfer coefficients to update the values of the film coefficients. If the newly calculated values are equal to the assumed values, the algorithm stops. Otherwise, the film heattransfer coefficients are updated and used in the next iteration until convergence is achieved. Two types of applications are shown. The first involves the solution of problems with a fixed energy recovery level (which implies a given value of ∆Tmin). The second type recognizes the fact that the total area and possibly the structure of the network will vary when the energy recovery value is changed; therefore, one can find the network with the minimum total (global) yearly cost through the calculation of the annual cost for several values of the energy recovery level, i.e., for different values of ∆Tmin. In the main body of the paper, the model equations are presented first, followed by a description of the synthesis algorithm. The paper concludes with the application of the method to three case studies taken from published works.
[
]
1 - Rdi fihi
-m
∀ i ) 1, 2, ..., NH, NC (3)
2.2. MINLP Model. The problem can be represented through a superstructure with stages that gives rise to an MINLP model2 whose solution should provide the network with the minimum yearly cost. Within the superstructure, the number of stages does not have to be equal to the number of energy intervals, and the temperatures for each stage are treated as optimization variables. In general, the number of stages required for the model will usually be lower than the number of hot streams, NH, or than the number of cold streams, NC. The superstructure is generated in the following way: 1. The number of stages is specified as max{NH, NC}. 2. For each stage, a stream is split to provide any possible exchange between hot and cold streams. The exit streams from each heat exchanger are mixed to provide the inlet stream to the next stage. 3. The outlet temperatures for each stage are treated as variables. Figure 2 shows a superstructure for two hot and two cold streams, along with heating and cooling utilities. The MINLP model can then be written as follows: Energy Balance for Each Stream. The heat transfer for each stream should be equal to the energy integrated with other process streams plus the energy given to process utilities
(TINi - TOUTi)Fi )
∑ ∑ qijk + qcui,
i ∈ HP
k∈STj∈CP
2. Model Formulation The main equations for the two-level optimization model are described in this section. 2.1. Heat Exchanger Model. The equations to account for pressure drops in heat exchangers are incorporated into the optimization task. A compact formulation for this purpose for any stream i can be written as6-8
∆Pi ) KiAcihcim
∀ i ) 1, 2, ..., NH, NC
(1)
where ∆Pi is the specified pressure drop, Ki is a constant that depends on the physical properties of stream i, Aci is the contact area (exchanger area), and hci is the clean film heat-transfer coefficient. The contact area Aci of stream i depends on the fouling film coefficients (hi and hj, see Figure 1). The clean and fouling film coefficients are related by
hci )
(
1
1 - Rdi fihi
)
∀ i ) 1, 2, ..., NH, NC
(2)
where Rdi is the specified fouling factor for stream i. The
(TOUTj - TINj)Fi )
(4)
∑ ∑ qijk + qhuj,
j ∈ CP
k∈STi∈HP
Energy Balance for Each Stage. An energy balance for each stage of the superstructure is required for the calculation of temperatures. For a superstructure of NOK stages there are NOK + 1 temperatures, as the outlet temperature from one stage is equal to the inlet temperature to the following one. The location of stage k ) 1 corresponds to the highest temperature value. Energy balances for each stage are as follows
(ti,k - ti,k+1)Fi )
∑ qijk,
k ∈ ST, i ∈ HP
j∈CP
(tj,k - tj,k+1)Fj )
∑ qijk,
(5) k ∈ ST, j ∈ CP
i∈HP
Specification of Inlet Temperatures to the Superstructure.
TINi ) ti,1, i ∈ HP TINj ) tj,NOK+1, j ∈ CP
(6)
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Temperature Feasibility.
min
ti,k g ti,k+1, k ∈ ST, i ∈ HP tj,k g tj,k+1 k ∈ ST, j ∈ CP TOUTi e ti,NOK+1, i ∈ HP
∑ CCU × qcu + ∑ CHU × qhu + i
i∈HP
∑ ∑ ∑ CF z
(7)
∑ ∑ ∑C
i∈HPj∈CPk∈ST
{[ {[
Heating and Cooling Duties.
(TOUTj - tj,1)Fj ) qhuj, j ∈ CP
(8)
∑C
Logical Restrictions. Logical restrictions and binary variables are used to determine the existence of an exchange between process streams (i,j) in stage k and some exchange with a process utility. The restrictions are represented as follows
(9)
zijk, zcui, zhuj ) {0, 1}
∑ CF
i∈HP
∑C
j∈CP
i,cuzcui
hu,jzhuj
}
+
( ) ( )] ( ) ( )] ( ) ( )] 1
qcui
1
1
+
hi
β
hj
dtijk + dtij,k+1
(dtijk)(dtij,k+1)
1
+
hi
hcu
dticu + TOUTi - TINcu
(dthuj)(TINhu - TOUTj)
+
1/3
2
qhuj
huj
∑ CF
+
j∈CP
qijk
(dticu)(TOUTi - TINcu)
qijk - Ωzijk e 0, i ∈ HP, j ∈ CP, k ∈ ST qhuj - Ωzhuj e 0, j ∈ CP
ij
+
icu
i∈HP
qcui - Ωzcui e 0, i ∈ HP
{[
ij ijk
i∈HPj∈CPk∈ST
TOUTj g tj,1, j ∈ CP
(ti,NOK+1 - TOUTi)Fi ) qcui, i ∈ HP
j
j∈CP
+
hhu
1
hj
dthuj + TINhu - TOUTj 2
+
1/3
2
1
}
β
}
β
1/3
(11)
3. Algorithm for the Synthesis of HENs
where the corresponding upper limit, Ω, is the smallest heat content of the two streams exchanging heat. Calculation of Temperature Differences. Because area requirements for each match are incorporated into the objective function, temperature differences need to be calculated. The synthesis model uses a pair of variables for the temperature differences dt. Binary variables are used to ensure that the temperature differences for each exchanger are feasible
The global MINLP model is solved through a decomposition strategy. For a given energy recovery level, the solution strategy consists of an iterative procedure to determine the optimum structure of a heat exchanger network that includes pressure drops for each stream. The search consists of the following steps: Step 1. Assumed values for foul heat-transfer coefand hguess ). ficients are initially used (hguess i j Step 2. The constant Ki is calculated with one of the following equations10
dtijk e ti,k - tj,k + Γ(1 - zijk), k∈ ST, i ∈ HP, j ∈ CP
φt4.5Dti1/2µt11/6
dtijk+1 e ti,k+1 - tj,k+1 + Γ(1 - zijk), k ∈ ST, i ∈ HP, j ∈ CP (10) dtcui e ti,NOK+1 - TOUTcu + Γ(1 - zcui), i ∈ HP dthuj e TOUThu - tj,1 + Γ(1 - zhuj), j ∈ CP where Γ is an upper limit for the temperature difference between process streams. These equations are written as inequalities because the cost of exchangers decreases with increasing temperature difference dt. Therefore, the role of the binary variables in eq 10 is to prevent negative temperature differences. When an exchange between streams (i,j) occurs in stage k, the variable zijk is equal to 1, and the restriction applies so that the temperature differences are calculated accordingly. Similar restrictions are written for the use of utilities to prevent the exit temperatures (TOUT) from becoming higher (for heating utilities) or lower (for cooling utilities) than the target temperatures for the process streams. Objective Function. The objective function is taken as the total yearly cost, given by the cost of the utilities and the fixed and variable costs of heat exchangers. Chen’s approximation9 is used for the calculation of the logarithmic mean temperature differences. The objective function is written as
Ki )
2.5
7/3
(0.023) gcMtFtkt Cp,t
7/6
( ) Dti Dt
(12)
if stream i flows through the tubes
(
)(
)(
)
67.062φs6.109CAT Ltp - Dt LtpDe1.109µs1.297 Ki ) gc Dt MsFsks3.406Cp,s1.703 (13) if stream i flows through the shell. Step 3. Using an MINLP model such as the one developed by Yee and Grossmann,2 we determine the HEN with the minimum annual cost. To solve the model, a combined penalty function and outer approximation method is used. The method involves the initial solution of a relaxed NLP, together with an MILP master problem with an augmented penalty function that accounts for violations of linearizations of the nonlinear functions. The penalty function uses weights equal to 1000 times the absolute magnitudes of the Karush-Kuhn-Tucker multipliers. The details of the method are available in Viswanathan and Grossmann.11 The algorithm was implemented in the DICOPT++ solver of the GAMS program.12 Step 4. Once the MINLP model is solved, the contact area for each stream is known, which allows for the
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Figure 2. Superstructure with two stages.
calculation of the fouling heat-transfer coefficients with the exchanger model. For stream i
[(
)
1 KPTiAci hi ) fi ∆PTi
1/3.5
]
-1
+ Rdi)
(14)
if stream i flows through the tubes
hi )
[(
)
KPTiAci ∆PTi
1/5.109
]
-1
+ Rdi)
(15)
if stream i flows through the shell. Step 5. Check convergence for each stream
- hguess | e |hguess | |hcalc i i i
(16)
where is a specified tolerance. If convergence is not achieved, one needs to update the assumed value of the film coefficient and go back to step 3. We used a simple ) hcalc direct substitution strategy, hguess i i , to fix the next values of the film coefficients with excellent results. The model used in this work has the advantage that the heat exchanger network is designed under the same basis as the detailed design of the exchangers, which is typically carried out later on the basis of allowable pressure drops for the streams. It should be noted that Frausto et al.13 reported an MINLP model in which the simultaneous optimization of energy, contact areas, and pressure drops was attempted. Such a formulation, however, has been successfully applied only to problems of small size (with few process streams). The algorithm presented in this work decomposes the problem into two levels as a strategy to solve rather complex problems; the resulting structure is simpler and much easier to solve. As a result, problems with a higher number of streams can now be solved. 4. Results and Discussion Three case studies are used to show the application of the proposed algorithm. In all cases, the convergence factor for eq 16 was taken as 1 × 10-6. Example 1. The data given in Tables 1 and 2 were taken from Shenoy.10 The problem is to find an HEN
Table 1. Stream Data for Example 1 stream
TIN (°C)
TOUT (°C)
F (kW/°C)
cost [$/(kW year)]
H1 H2 ST C1 C2 CW
175 125 180 20 40 15
45 65 179 155 112 25
10 40 20 15 -
110 10
Table 2. Physical Properties and Allowable Pressure Drops for Example 1 ∆P (kPa) Rd [(m2 °C)/W] Cp [J/(kg °C)] F (kg/m3) µ [kg/(m s)] K [W/(m °C)]
H1
H2
C1
C2
5.27 0.00015 1658 716 0.24 × 10-3 1.1
11.19 0.00015 2684 777 0.23 × 10-3 0.24
10 0.00015 2456 700 0.23 × 10-3 0.12
10 0.00015 2270 680 0.23 × 10-3 0.011
Table 3. Initial Values of Film Coefficients for Example 1 stream
h [kW/(m2 °C)]
H1 H2 C1 C2
2.489 1.319 0.903 0.158
with the minimum yearly cost for ∆Tmin ) 20 °C. Pressure drop values for each stream were taken from Frausto et al.13 The cost function for heat exchangers is $30,800 + $890(A)0.8 (area in m2). The outer and inner diameters for the tubes are 19.1 and 15.4 mm, respectively. The tubes are placed in a square arrangement with a 25.4mm spacing. Hot streams flow through the shell sides. Film heat-transfer coefficients for utilities are assumed constant and equal to 5 kW/(m2 °C) for steam and 2.5 kW/(m2 °C) for cooling water. The initial values of the film heat-transfer coefficients for process streams are given in Table 3; these values were taken from Polley and Panjeh Shahi.4 The proposed algorithm took seven iterations to find the optimum solution. The MINLP model consisted of a superstructure that included 12 binary variables. The optimum network is shown in Figure 3 and consists of seven units (four internal exchangers, one cooler, and
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Figure 3. Optimum network design for example 1. Table 4. Comparison of Results for Example 1 Frausto et total area (m2) heating utility consumption (kW) cooling utility consumption (kW) number of units total cost ($/year)
al.13
486.96 605 525 7 444,714.93
Table 6. Stream Data for Example 2 this work
stream
TIN (°C)
TOUT (°C)
F (kW/°C)
cost [$/(kW year)]
429.49 605 525 7 439,535.88
H1 H2 ST C1 C2 CW
150 90 180 20 25 10
60 60 180 125 100 15
20 80 25 30 -
110 10
Table 5. Film Coefficients Obtained for Example 1 stream
h [kW/(m2 °C)]
H1 H2 C1 C2
2.084 1.173 0.845 0.157
two heaters) with a total area of 429.49 m2. The annual investment cost is $367,735/year. The optimum structure determined by the algorithm is such that stream H1, the hot stream with the highest film coefficient, exchanges heat with stream C1, the cold stream with the highest film coefficient; in principle, such an arrangement should optimize the use of the heat-transfer area. Two stream splits are needed for the optimum network. For the specified energy recovery level, the total annual cost is then $439,535/year. Table 4 compares our solution with that reported by Frausto et al.13 Both methods use the same optimization code DICOPT++ from the GAMS software (which implies that a global optimum solution cannot be guaranteed in either case because of the model nonconvexities), but the two-level decomposition strategy used in this work leads to a better solution than that obtained with the simultaneous approach employed by Frausto et al. for the same allowable pressure drops for the streams. Table 5 reports the fouling heat-transfer coefficients that were obtained for each stream. Example 2. This case study was taken from Polley and Panjeh Shahi.4 Tables 6 and 7 list the design data for the streams. The same economic and geometric data as in example 1 were used. Pressure drop values for streams were taken from Frausto et al.13 The energy recovery level corresponds again to the use of ∆Tmin ) 20 °C. The film heat-transfer coefficients for both heating and cooling utilities were assumed equal to 0.1 kW/(m2 °C). For process streams, the values for the film coefficients used by Polley and Panjeh Shahi4 of 0.1 kW/ (m2 °C) for all streams were used to initialize the algorithm.
Table 7. Physical Properties and Allowable Pressure Drops for Example 2 ∆P (kPa) Rd [(m2 °C)/W] Cp [J/(kg °C)] F (kg/m3) µ (cPs) K [W/(m °C)]
H1
H2
C1
C2
20 0.00018 2600 800 0.5 0.12
9.98 0.00018 2600 800 0.5 0.12
6.71 0.00018 2600 800 0.5 0.12
23.16 0.00018 2600 800 0.5 0.12
A superstructure with two stages that imbedded 12 binary variables was constructed for the MINLP model. The use of the proposed algorithm to determine the optimum solution took seven iterations, and the optimum network is shown in Figure 4. Three stream splits are observed within the optimum network structure. The total area required for the seven heat-transfer units of the network amounts to 577.48 m2, with an annual cost of $237,407/year. For the specified energy recovery level, the heating and cooling requirements are 1,075 and 400 kW, respectively. The total annual cost for the network is thus $359,657/year. The optimum network determined here was based on the same energy recovery level and the same allowable pressure drops for each stream as used by Frausto et al.13 A comparison of the results between these two works is given in Table 8. The two solutions show a similar configuration, but the design obtained with the proposed algorithm requires 40.76 m2 less area (which amounts to a 7% reduction with respect to the solution of Frausto et al.). As in the case of the first example, it seems that the decomposition strategy of the method presented in this work might lead to a better search toward a global (or at least a better local) optimum solution than the simultaneous formulation by Frausto et al. The fouling heat-transfer coefficients obtained after the application of the algorithm are reported in Table 9.
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Figure 4. Optimum solution for example 2. Table 8. Results Comparison for Example 2 total area (m2) heating utility requirements (kW) cooling utility requirements (kW) number of units total cost ($/year)
Frausto et al.13
this work
618.24 1,075 400 7 372,187.03
577.48 1,075 400 7 359,657.33
Table 9. Film Coefficients Obtained for Example 2 stream
h [kW/(m2 °C)]
H1 H2 C1 C2
0.698 0.764 0.481 0.794
Table 10. Stream Data for the Aromatics Plant stream
TIN (°C)
TOUT (°C)
F (kW/°C)
cost [$/(kW year)]
H1 H2 H3 H4 oil C1 C2 C3 C4 C5 CW
327 220 220 160 330 100 35 85 60 140 10
40 160 60 45 230 300 164 138 170 300 30
100 160 60 400 100 70 350 60 200 -
80 15
Example 3. We now consider the fairly well-known problem of the aromatics plant, which has been considered by several authors.4,14 The problem consists of four hot streams and five cold streams, along with heating and cooling utilities. The data on stream temperatures, physical properties, and assumed allowable pressure drops for each stream are given in Tables 10 and 11. The film heat-transfer coefficients for the utilities were taken as constant and equal to 1 kW/(m2 °C) for heating oil and 2.5 kW/(m2 °C) for cooling water. Countercurrent shell and tube heat exchangers are considered. The tubes are placed in a square arrangement with a spacing of 25.4 mm and have outer and inner diameters of 19.1 and 15.4 mm, respectively. Hot streams are assumed to flow through the shell side, and the cost function for the heat exchangers is taken as $5,500 + $150(A), with A in m2. The aromatics problem has only been solved using the principles of the pinch-point method. Regarding math-
Table 11. Physical Properties and Pressure Drop Data for the Aromatics Plant Cp ∆P F µ K Rd stream (kg/m3) [J/(kg °C)] (cPs) [W/(m °C)] [(m2 °C)/W] (kPa) H1 H2 H3 H4 C1 C2 C3 C4 C5
500 55 676 679 464 570 1 685 667
2000 2192 1877 5480 2000 1590 10000 1580 2740
0.25 0.01 0.28 0.31 0.16 0.30 0.01 0.27 0.21
0.11 0.026 0.11 0.11 0.11 0.11 0.17 0.11 0.11
0.00018 0.00018 0.00018 0.00018 0.00018 0.00018 0.00018 0.00018 0.00018
120 80 90 60 20 20 30 15 80
ematical programming techniques, it should be noted that the simultaneous MINLP model by Frausto et al.13 failed to provide a solution for this problem. Therefore, this example provides a good test for the proposed algorithm, as the number of streams involved in this case typically creates significant convergence problems for the solution of MINLP models. Furthermore, we carried out a recursive use of the model for several energy recovery levels to detect the value of ∆Tmin that provided the network with the global minimum cost. For each energy recovery level, a superstructure with five stages was required for the aromatics problem, with 109 potential matches. The MINLP model included 109 binary variables and 273 continuous variables. The search space for ∆Tmin consisted of six values. The first point to highlight from the solution process is that the proposed algorithm achieved convergence for all of the search points considered. The decomposition strategy for the solution of the MINLP model, therefore, shows a great potential for the solution of large syn-
Figure 5. Optimization of the energy recovery level for the aromatics problem.
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Figure 6. Optimum network obtained for example 3. Table 12. Initial and Final Film Coefficients for Example 3 [kW/(m2 °C)] stream
initial h
final h
H1 H2 H3 H4 C1 C2 C3 C4 C5
0.797 0.576 0.853 0.966 0.815 0.710 1.344 0.736 1.208
1.089 0.738 0.925 0.924 0.931 0.793 1.514 1.064 1.341
thesis problems. For most of the search points, the algorithm took seven iterations to converge. Figure 5 shows the objective function value obtained after the model was applied recursively for the six values of ∆Tmin. The global minimum yearly cost was obtained for ∆Tmin ) 18 °C. Figure 6 shows the optimum HEN obtained for the best value of ∆Tmin of 18 °C. The initial and final values of film heat-transfer coefficients for this case are reported in Table 12. The network consists of 11 heat exchangers, with a total area of 4,442 m2 (for an investment required of $1,127,287) and a total annual cost of $5,784,087. No stream splits are needed for the optimum structure. Although there is no clear trend in terms of the matches for the optimum solution, one can observed that the cold stream with the highest film coefficient (C3) is matched with the two hot streams with the lowest film coefficients (H2 and H4). In any event, it is the algorithm’s task to provide the proper matches for the best energy-heat exchanger-pumping equipment compromise. The successful application of the algorithm to this case study shows an interesting potential of the proposed method to solve medium- or large-size synthesis problems. 5. Conclusions The incorporation of allowable pressure drop values within the formulation of the synthesis of a heat exchanger network conveys a problem of significant importance for both industrial and technical purposes. The first advantage of such a formulation is that it eliminates the subjective values of film coefficients
typically given as design data for a heat exchanger network problem. The second advantage is that one carries out both the synthesis of the network and the detailed equipment design on a consistent basis, as both are based on the allowable pressure drops for the streams. An optimization algorithm for the synthesis of HENs that includes pressure drop considerations has been presented. For a given heat recovery level, the algorithm uses a search procedure on two levels. In an inner loop, an MINLP model such as the one suggested by Yee and Grossmann2 is used with a set of assumed values for the film coefficients to detect an optimum network structure. In the outer level, the solution of the MINLP model provides the contact areas required for each process stream and, therefore, allows for the calculation of new values of the film heat-transfer coefficients as a function of the allowable pressure drop values. The values of assumed and calculated film heat-transfer coefficients are used in each iteration to check convergence. The model is suitable for cases with significant differences in the physical properties of the streams, such as their film heat-transfer coefficients. Constraints on certain stream matches (due to safety or location aspects, for example) can be readily incorporated into the method; such a case is of particular importance for HEN retrofit problems. The decomposition of the main problem into two loops provides an algorithm with excellent convergence properties that are not typically observed with the use of an overall MINLP formulation such as the one used by Frausto et al.13 Nomenclature Ac ) contact area C ) area cost coefficient CCU ) unit cost for cold utility CHU ) unit cost for hot utility CF ) ixed charge for exchangers CP ) {j|j is a cold process stream} Cp ) heat capacity CU ) cold utility De ) diameter (equivalent) Dti ) diameter (inner) of tube
Ind. Eng. Chem. Res., Vol. 43, No. 21, 2004 6773 dtijk ) temperature approach for match (i,j) at temperature location k dtcui ) temperature approach for match of hot stream i and cold utility dthuj ) temperature approach for match of cold stream j and hot utility f ) dimensionless factor F ) heat capacity flow rate h ) fouling heat-transfer coefficient HP ) {i|i is a hot process stream} HU ) hot utility k ) thermal conductivity KPS ) pressure drop coefficient for the shell-side fluid KPT ) pressure drop coefficient for the tube-side fluid Ltp ) tube pitch M ) mass flow rate of fluid streams NOK ) total numbers of stages ∆P ) allowable pressure drop qij ) heat exchanged between opposites process stream i and j qijk ) heat exchanged between hot process stream i and cold process stream j in stage k qchj ) heat exchanged between hot utility and cold stream j qcui ) heat exchanged between cold utility and hot stream i QCU ) total cold utility QHU ) total hot utility Rd ) fouling resistance factor 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 ∆Tmin ) minimum approach temperature difference TINi ) inlet temperature of stream i TOUTi ) outlet temperature of stream i U ) overall heat-transfer coefficient zijk ) binary variable for match (i,j) in stage k zcui ) binary variable for match between cold utility and hot stream i zhuj ) binary variable for match between hot utility and cold stream j Greek Symbols β ) exponent for area cost equation ) convergence criterion Ω ) upper bound for heat exchange F ) density Γ ) upper bound for temperature difference µ ) viscosity
Subscripts i ) hot process stream j ) cold process stream k ) index for stage (1, ..., NOK) and temperature location (1, ..., NOK + 1) s ) shell t ) tube
Literature Cited (1) Linnhoff, B.; Hindmarsh, E. The pinch design method of heat exchanger networks. Chem. Eng. Sci. 1983, 38, 745-763. (2) Yee, T. F.; Grossmann, I. E. Simultaneous Optimization Models for Heat IntegrationsII Heat Exchanger Network Synthesis. Comput. Chem. Eng. 1990, 14, 1165-1184. (3) Polley, G. T.; Panjeh Shahi, M. H.; Jegede, F. O. Pressure drop considerations in the retrofit of heat exchanger networks. Trans. Inst. Chem. Eng. A 1990, 68, 211-220. (4) Polley, G. T.; Panjeh Shahi, M. H. Interfacing heat exchanger network synthesis and detailed heat exchanger design. Trans. Inst. Chem. Eng. A 1991, 69, 445-457. (5) Townsend, D. W.; Linnhoff, B. Surface area targets for heat exchanger networks. Presented at the IChemE 11th Annual Research Meeting, Bath, U.K., April 1984. (6) Peters, M. S.; Timmerhaus, K. D. Plant Design and Economics for Chemical Engineers, 4th ed.; McGraw-Hill: Singapore, 1991. (7) Jegede, F. O.; Polley, G. T. Optimum heat exchanger design. Trans. Inst. Chem. Eng. 1992, 70, 133-141. (8) Serna, G. M. Desarrollo de Algoritmos Rigurosos para la Integracio´n Te´rmica de Procesos. Ph.D. Thesis. Instituto Tecnolo´gico de Celaya, Celaya, Me´xico, 1999. (9) Chen, J. J. J. Comments on improvements on a replacement for the logarithmic mean. Chem. Eng. Sci. 1987, 42, 2488-2489. (10) Shenoy, U. V. Heat Exchanger Network. Process Optimization by Energy and Resource Analysis; Gulf Publishing Company: Houston, TX, 1995. (11) Viswanathan, J.; Grossmann, I. E. A Combined Penalty Function and Outer Approximation Method for MINLP Optimization. Comput. Chem. Eng. 1990, 14, 769-782. (12) Brooke, A.; Kendrick, M. A.; Raman, R. GAMS-A User’s Guide; GAMS Development Corporation: Washington, DC, 1988. (13) Frausto, S. H.; Rico-Ramı´rez, V.; Jime´nez-Gutie´rrez, A.; Herna´ndez-Castro, S. MINLP synthesis of heat exchanger networks considering pressure drop effects. Comput. Chem. Eng. 2003, 27, 1143-1152. (14) Tjoe, T. N.; Linnhoff, B. Using pinch technology for process retrofit. Chem. Eng. 1986, April 28, 47-60.
Received for review March 23, 2004 Revised manuscript received July 9, 2004 Accepted August 13, 2004 IE0497700
