Improvement on the Simultaneous Optimization Approach for Heat

Article pubs.acs.org/IECR

Improvement on the Simultaneous Optimization Approach for Heat Exchanger Network Synthesis Guoqing Li,* Yushu Luo, Yong Xia, and Ben Hua †

South China University of Technology, Tianhe, Guangzhou 510640, People’s Republic of China ABSTRACT: In this article, three corrections are applied to improve the simultaneous optimization synthesis approach (SOSA) of heat exchanger networks (HENs). First, the temperature-dependent characteristics of process streams are considered in calculating their heat capacities, which are regarded as constant within a defined temperature interval; second, splitters and mixers of streams located in the network are listed as investment items, resulting in more detailed HEN cost calculations; third, a group of heuristic rules is suggested to cancel superstructure matches that are impossible in terms of engineering or thermodynamics before the optimization calculation, making the simultaneous optimizations on both investment cost and utility consumption charge available while also saving significant computation time. Based on these corrections, an improved SOSA is developed in which an MINLP model is solved by the use of genetic algorithms to avoid being trapped in a local optimum. A case study shows that this approach realizes significant progress in both precision and computation-time savings.

1. INTRODUCTION Heat exchanger network (HEN) synthesis is one of the moststudied problems in chemical engineering.1 HEN synthesis methods can be divided into two main categories: pinch analysis techniques and mathematical programming techniques.2 In the past few decades, pinch analysis techniques have played a main role in the synthesis of industrial HENs.3 However, mathematical programming techniques have also received increasing attention in recent years with the rapid development of computer technology.2 In mathematical programming techniques, the problem of HEN synthesis is formulated as a mixed-integer nonlinear programming (MINLP) problem:4 the best solution satisfying the specified target is selected from the superstructure assembly that contains all possible matches. Because the target is usually defined as the minimum total annual cost of the HEN, including running charges (e.g., utility consumption) and capital investment depreciation, the procedure is called a simultaneous optimization synthesis approach (SOSA).5 Numerous investigations have been carried out to improve this SOSA. Yee and Grossmann employed a stagewise superstructure (see Figure 1) for HEN design. In their initial works,4 they took the utility consumption and the area cost of heat exchangers into consideration but not the fixed cost of heat exchangers. Soon, a modified model was developed, but it was based on the assumption of isothermal mixing, according to which all streams that enter the same mixer must be at the same temperature.5 To overcome this unreasonable assumption, Yuan and Ying6 suggested an improved MINLP transshipment model, but it requires that the heat capacities of all process streams that participate in heat exchange must be constant and also does not include the charge of any splitter or mixer located in the HEN. In fact, nonconstant thermal properties often arise, especially when the temperature varies in a large range. However, only a few methodologies consider varying thermal properties (e.g., heat capacities) of process streams. Smith et al.2 broke the © 2012 American Chemical Society

Figure 1. HEN superstructure proposed by Yee and Grossmann.5

temperature range of a stream into several subintervals but still assumed that the specific heat of the stream remained constant in each subinterval. This approach can decrease the error but provides no essential improvement. A superstructure externalizes very well the notion of mathematical programming techniques in the synthesis of HENs, but calculating the corresponding MINLP model is very difficult and even impossible when more streams are involved in a network. Kravanja and Glavič7 combined composite curves3 with superstructure study, defining the number of stages of the superstructure to be equal to the number of enthalpy intervals in the composite curve and also introducing the assignment coefficient to denote the existence of matches at each stage for each pair of hot and cold streams. Because most of the assignment coefficients are zero, the size of the Received: Revised: Accepted: Published: 6455

October 4, 2011 March 6, 2012 April 6, 2012 April 6, 2012 dx.doi.org/10.1021/ie202271h | Ind. Eng. Chem. Res. 2012, 51, 6455−6460

Industrial & Engineering Chemistry Research

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cpp = a0p + a1pT + a 2pT 2 + a3pT 3

superstructure is reduced significantly. However, no explanation of how to select the assignment coefficient was proposed in their work.

where a0p, a1p, a2p, and a3p are coefficients associated with the heat capacity of the pth process stream involved in the HEN. They can be conveniently obtained from some manuals on stream properties. For all n process streams involved in the HEN, one can write eq 2 as the matrix product

2. IMPROVED SIMULTANEOUS OPTIMIZATION SYNTHESIS APPROACH ON HEN In this section, an improved approach for the simultaneous optimization synthesis of HENs is presented, in which (1) the costs of all splitters and mixers located in the network are included, (2) the heat capacities of all process streams are calculated from a strict temperature-dependent relationship, (3) a group of heuristic rules is suggested to simplify the HEN superstructure before the optimization calculation, and (4) an optimization calculation method based on genetic algorithms is suggested for solving the improved MINLP model. 2.1. Stagewise Superstructure. The representation of the HEN structure in this work employs a stagewise superstructure as shown in Figure 1. The HEN is modeled as an interconnected set of network elements, namely, process heat exchangers, utility heat exchangers, stream splitters, and mixers. Generally, the number of stages (Nk) is set equal to the maximum cardinality of the hot and cold sets of streams Nk = max{Nh, Nc}

(2)

cp = a × T

(3)

where cp = (cp1, cp2 , ..., cpn)T

(4)

⎡ a01 a11 a 21 a31 ⎤ ⎢a a a a ⎥ 02 12 22 32 ⎥ a=⎢ ⎢⋮ ⋮ ⋮ ⋮⎥ ⎢ ⎥ ⎣ a0n a1n a 2n a3n ⎦

(5)

T = (1, T , T 2 , T 3)T

(6)

To calculate the heat capacities of process streams, one can use the equation

(1)

CP = cp × F

At each stage, hot and cold streams are split to allow the potential existence of a heat exchanger to match any hot−cold pair of streams. Auxiliary cooling and heating utilities are placed at the outlets of the superstructure. 2.2. Heat Capacity Representation. In Figure 2, a cold stream is heated from T1 to T2 through two heat exchangers,

(7)

with

F = (F1 , F2 , ..., F n)T

(8)

where Fp is flow rate of the pth process stream. 2.3. Heat Exchanger Network Model. The principal elements for modeling a HEN are described in the following subsections. Stream Inlet Temperatures. The initial temperature of each stream is its supply temperature to the superstructure Ti ,in = Ti ,S

i ∈ Nh

(9)

Tj ,in = Tj ,S

j ∈ Nc

(10)

Stream Outlet Temperatures. The outlet temperature of each stream is its temperature when leaving the superstructure Ti ,out = Ti ,Nk

i ∈ Nh

Tj ,out = Tj ,1

(11)

j ∈ Nc

(12)

Heat Duty of Heat Exchanger. Because of the nonconstant thermal properties of streams, an integral should be employed for the calculation of the heat duty of a heat exchanger

Figure 2. Demonstration of heat capacity of streams.

Q ijk =

labeled 1 and 2, in two different forms: keeping the heat capacity constant (form 1) and varying the heat capacity with the temperature (form 2). It is evident that the heat-transfer area, AC1, of heat exchanger 1 in form 1 is not equal to AN1 (the area of heat exchanger 1 in form 2); at the same time, AC2 also not equal to AN2 (the area of heat exchanger 2 calculated in the form 2) because of the different intermediate temperatures TC3 and TN3 in forms 1 and 2, respectively. This indicates that different means of handling the heat capacities of process streams generate different optimization results in the synthesis of HENs. Thus, to describe real thermal properties of process streams, use of nonconstant specific heats is necessary.2 Equation 2 can be used to identified the correlation between heat capacity (cp) and temperature8

∫T

Ti , k

CPd i T

i ∈ Nh, k ∈ Nk

CPd j T

j ∈ Nc, k ∈ Nk

i ,k+1

Q ijk =

∫T

Tj , k

(14)

j,k+1

Q i ,cu =

Ti ,out

∫T

CPd i T

i ∈ Nh

CPd j T

j ∈ Nc

i ,T

Q j ,hu =

∫T

Tj ,T

j ,out

(13)

(15)

(16)

where Ti,T and Tj,T are the target temperatures of hot stream i and cold stream j, respectively. Substituting eq 2 into eqs 13−16 gives 6456

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Industrial & Engineering Chemistry Research Q ijk = yijk

∫T

Ti , k

difference, and dthijk and dtcijk are the temperature approaches of hot side and cold sides, respectively. Constraints. The temperature approach of heat exchanger is controlled above the exchanger minimum approach temperature (EMAT) boundary

(a0i + a1iT + a 2iT 2 + a3iT 3)dT

i ,k+1

i ∈ Nh, k ∈ Nk

Q ijk = yijk

Article

∫T

Tj , k

(17)

(a0j + a1jT + a 2jT 2 + a3jT 3)dT

dthijk = Ti , k − Tj , k − ΔTmin ≥0

j,k+1

j ∈ Nc, k ∈ Nk

∫T

Q i ,cu = yi ,cu

Ti ,out

i ∈ Nh, j ∈ Nc, k ∈ Nk

(18)

dtcijk = Ti , k + 1 − Tj , k + 1 − ΔTmin ≥0

(a0i + a1iT + a 2iT 2 + a3iT 3)dT

i ,T

i ∈ Nh, j ∈ Nc, k ∈ Nk (19)

i ∈ Nh

Q j ,hu = yj ,hu

∫T

Tj ,out

2

(a0j + a1jT + a 2jT + a3jT )dT

Q min ≤ Q ijk ≤ Q max

j,T

(20)

min Γ =

∑ +

(cuQ i ,cu + βAai ,cu )

∑

(huQ j ,hu + γAa j ,hu )

j ∈ Nc

+

∑ ∑ ∑

[αAaijk + yijk PHijk

i ∈ Nh j ∈ Nc k ∈ Nk

+ zijk(PSijk + PMijk)]

(21)

⎛1 ⎞ 1 1 yijk ⎜ a3jTj , k 4 + a 2jTj , k 3 + a1jTj , k 2 + a0jTj , k ⎟ − Cj , k ⎝4 ⎠ 3 2 j ∈ Nc, k ∈ Nk

(29)

i ∈ Nh

⎛1 ⎞ 1 1 yijk ⎜ a3iTi , k 4 + a 2iTi , k 3 + a1iTi , k 2 + a0iTi , k ⎟ − Ci , k = 0 ⎝4 ⎠ 3 2

=0

i ∈ Nh, j ∈ Nc, k ∈ Nk

Objective Function. The objective function is defined as

where yijk, yi,cu, and yj,hu are Boolean variables equal to 0 or 1 that represent the existence of the process exchanger ijk and the cold and hot utilities, respectively. Intermediate Temperatures. The intermediate temperatures Ti,k and Tj,k can be calculated by solving the equations

i ∈ Nh, k ∈ Nk

(28)

The heat duty of a heat exchanger should be bounded within a certain range

3

j ∈ Nc

(27)

(30)

In addition to the costs of utility consumption and heat exchanger investment (including area-dependent costs and fixed costs), the costs of all splitters and mixers located in the network are also taken into account. 2.4. Overall Approach. HEN synthesis based on a superstructure is a very complicated MINLP problem, especially when many streams are involved. Therefore, the general optimization procedure is usually divided into two steps,2,9−11 as shown in Figure 3. First, a single-target optimization of minimizing the total investment is carried out

(22)

where ⎛1 1 1 Ci , k = Q ijk + yijk ⎜ a3iTi , k + 14 + a 2iTi , k + 13 + a1iTi , k + 12 ⎝4 3 2 ⎞ + a0iTi , k + 1⎟ ⎠ ⎛1 1 1 Cj , k = Q ijk + yijk ⎜ a3jTj , k + 14 + a 2jTj , k + 13 + a1jTj , k + 12 ⎝4 3 2 ⎞ ⎟ + a0jTj , k + 1 ⎠

Heat-Exchanger Area. After the inlet and outlet temperatures of a heat exchanger are defined, the required heattransfer area can be obtained as Aijk = Q ijk /(Uijk ΔTLM, ijk)

i ∈ Nh, j ∈ Nc, k ∈ Nk (23)

Ai ,cu = Q i ,cu/(Ui ,cuΔTLM, i ,cu)

i ∈ Nh

(24)

Aj ,hu = Q j ,hu/(Uj ,huΔTLM, j ,hu)

j ∈ Nc

(25)

where ΔTLM, ijk = (dthijk − dtcijk)/ln(dthijk /dtcijk) i ∈ Nh, j ∈ Nc, k ∈ Nk

(26)

The variable U represents the overall heat transfer coefficient, the subscript LM is short for logarithmic mean temperature

Figure 3. General two-step method for optimizing HENs. 6457

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duties of all hot streams and all cold streams, respectively; and CPi,sum and CPj,sum as the total heat capacity flow rates of all hot streams and all cold streams, respectively, these rules can be generalized as follows: (1) If the supply temperature of a hot stream is not higher than that of cold stream by the exchanger minimum approach temperature, then the streams are forbidden to match. (2) If Qi/Qi,sum ≤ 1/(ω1,iNh), then hot stream i should be arranged in only one or two stages rather than every stage. (3) If Qj/Qj,sum ≤ 1/(ω1,jNc), then cold stream j should be arranged in only one or two stages rather than every stage. (4) If CPi/CPi,sum ≤ 1/(ω2,iNh), then hot stream i does not need to be split to match with every cold stream and might even not need to be split. (5) If CPj/CPj,sum ≤ 1/(ω2,iNc), then hot stream j does not need to be split to match with every hot stream and might even not need to be split. (6) If a stream has a temperature range of less than 40 °C and a small heat capacity flow rate, then the stream can be allocated in only one stage. (7) Some other engineering constraints should also be considered. For example, if there is a long distance between two streams, then the match between them should be prohibited. Likewise, if there is a large pressure difference between a hot stream and a cold one, then the streams should not be matched. In addition, if a hot stream is a target product, then it should not be matched with corrosive cold streams so as not to be polluted. In these rules, ω1,i, ω2,i, ω1,j, and ω2,j are coefficients defined in the space interval of [1, 2] that can be chosen according to the concrete matching cases. Many case studies have shown that user interactions can pick out almost all matches that are impossible in terms of engineering and thermodynamics, thereby reducing the size of the superstructure significantly. Mathematical Optimization. Because temperature-dependent thermal properties are taken into consideration in this article, the nonlinearity of the model is increased markedly, and deterministic methods might fail to attain an optimum.2 In view that stochastic methods can provide more chances to find the optimum for MINLP problems,2,12,13 this article employs a genetic algorithm (GA). Dipama and Teyssedou14 pointed out that some classical deterministic methods, such as simplex, cannot solve nonlinear problems and that, when dealing with large constrained problems, classical methods must manipulate several high-dimensional tables, thus necessitating both high computational power and a large memory space. In contrast, GAs can handle any type of problem, regardless of whether they are linear. GAs have the following additional advantages: • Genetic algorithms regard MINLP problems as black boxes, so no derivatives are required and only the fitness function value of the trial solution is needed. • Genetic algorithms can powerfully avoid being trapped in local optima.15 • Genetic algorithms can handle more constraints, which is appropriate for HEN design. In this article, a continuous GA is applied to optimize HEN synthesis. The Boolean 0−1 integer variables representing the existence of exchangers can be converted by the equation

in the inner loop using inputs of a group of initial values for random selected variables such as split ratios and intermediate temperatures. Then, another single-target optimization of minimizing the total utility consumption is performed in the outer loop based on the output of the inner loop. It is evident that these two optimizations are not simultaneous, so that the solution can easily fall into a local optimum. To overcome the deficiency of this two-step procedure, this article suggests the one-step optimization procedure shown in Figure 4 in which the two optimizations of utility consumption

Figure 4. Improved one-step method for optimizing HENs.

and investment cost are performed simultaneously after addition of a user interaction stage. A detailed descriptiong of this stage is provided in the next section. User Interaction. The MINLP focuses on seeking the most cost-effective structure from among many candidates. The complexity of resolving the MINLP model in HEN synthesis depends on two main factors: the size of the superstructure and the nonlinearity of calculations such as those for heat-transfer area, temperature difference of heat transfer, and thermal heat properties. The latter factor exists objectively in any heattransfer process, as such calculations are very difficult to linearize. The former is defined by the number of all possible matches of hot and cold streams. The larger the number of matches, the larger the size of the superstructure and, accordingly, the larger the calculation scale of the MINLP problem. Hence, it is necessary to cancel the matches that can arise in the mathematical superstructure but are unreasonable in terms of engineering and thermodynamics before performing the optimization calculation. For example, some streams have relatively small heat capacity flow rates, meaning that there is no need for them to be split into several substreams and matched with other cold or hot streams. Similarly, the temperature ranges of some streams are relatively narrow, so the streams have to be arranged in only some specific stages rather than every stage. Moreover, the heat duties of some streams are smaller, meaning that there is no need to arrange for a utility exchanger. These potential rules can help greatly simplify the model. Defining Qi,sum and Qj,sum as the total heat 6458

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Industrial & Engineering Chemistry Research ⎧1 yijk ≥ 0.5 nijk = ⎨ ⎩ 0 otherwise

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Table 1. Stream Data for HEN Designa

⎪

⎪

(31) stream

where yijk ∈ [0, 1]. The population of the GA in this work consists of all variables involved in HEN design. In each given population, for any heat exchanger, parameters such as inlet temperature, outlet temperature, heat-exchange area, heat load, and investment cost can be calculated. If these parameters satisfy the constraints stipulated in eqs 27−29, the total annual cost is evaluated; otherwise, the candidate is abandoned, and the penalty function is assigned. A fitness function, which is the evaluation function of the genetic algorithm, is then calculated as fitness = A /Γ + penalty

H1 H2 H3 H4 H5 C1 C2 C3 C4 C5 stream cooling water

(32)

where Γ is the value of the objective function and A is a constant. The penalty term is given by

Tin (°C)

Tout (°C)

heat duty (kW)

correlation between the heat-capacity flow rate and temperature (kW/°C)

160 249 271 227 199 82 93 38 60 116 236 38

93 138 149 66 66 177 205 221 160 222 236 82

588.9 1171.1 1532.3 2378 2358.1 1641.6 1556.8 1544.5 762 644.5 − −

0.015T + 6.8925 0.022T + 6.2935 0.024T + 7.5198 0.02T + 11.8402 0.018T + 15.3451 0.0165T + 15.1432 0.019T + 11.069 0.02T + 5.85 0.015T + 5.97 0.021T + 2.531 − −

Steam utility cost = 37.64 US$ kW−1 year−1. Cooling water cost = 18.12 US$ kW−1 year−1. Overall heat-transfer coefficient (U) = 0.852 kW m−2 °C−1 for all matches except those involving steam. Overall heat-transfer coefficient (U) = 1.136 kW m−2 K−1 for matches involving steam. Minimum approach temperature (ΔTmin) = 10 K. Annual cost (US$ year−1) = 145.63A0.6 for all exchangers, where A is the area in m2. a

⎧0 solution is feasible penalty = ⎨ ⎩−inf otherwise

where inf is a very large number. Because the GA selects the maximum of the target function, the fitness function should be the inverse of the objective function. If the maximum number of generations is reached or if there is no improvement in the fitness function in 20 successive generations, then convergence of the algorithm is achieved. Otherwise, current individuals are randomly selected as the parents of a new generation, with higher weights on individuals with lower costs. The new generation is obtained using the genetic operators of crossover (merging two individuals) and mutation (randomly modifying one individual). The procedure is repeated until the convergence criterion (no improvement in two successive generations) is satisfied.10

3. CASE STUDY As an application, a published case16−19 with a 10-stream problem (five hot streams and five cold streams) is considered. The data for the 10 streams, including their supply and target temperatures and heat duties, are listed in the Table 1. To consider temperature-dependent characteristics of the streams in calculating their heat capacities, which are regarded as constant values in refs 16−19, this article considers the 10 streams to be 10 equivalent petroleum distillates, each keeping the original supply and target temperatures and heat duties, and builds the correlations between the CP and T by regressing ASPEN PLUS simulation results, also listed in Table 1. According to the procedure shown in Figure 4, the matches that are impossible in terms of engineering and thermodynamics should be canceled in advance in the user interaction stage. As can be seen in Table 1, the heat duty of stream H1 is relatively small, it should be arranged in only one stage rather than all stages according to rule 2. The heat duties of stream C4 and C5 are also small, so a one-stage arrangement is applied according to rule 3. Finally, based on rules 4 and 5, streams H1 and C3−C5 do not need to be split. The optimal HEN is shown in Figure 5, with seven heat exchangers and three coolers and a total annual cost of US $43,602. The comparison listed in Table 2 shows that the optimum HEN suggested in this article is more effective than those reported in refs 16−19 as follows: (1) The total annual cost is less than those in refs 17 and 18. (2) The number of

Figure 5. Optimum HEN structure suggested in this article.

Table 2. Comparison between the Optimum HEN Obtained in this Article and Others case this article ref 16 ref 17 ref 18 ref 19

total annual charge (US$)

number of heat exchangers

number of coolers

CPU time (s)

43,602

7

3

6842

43,439 43,934 43,799 43,329

7 − 8 8

3 − 3 2

10012.3 − − 28207.1

heat exchangers in the HEN is less than those in refs 18 and 19. (3) The CPU time taken in the calculation (Windows XP, PC with P4, 2.93 GHz, 512 MB of RAM) is less than those in refs 16 and 19. (4) The temperature distribution is closer to the real state than those in the other works because of the realistic description of heat capacity as a function of temperature, as confirmed by simulation. 6459

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(13) Gadalla, M. A. Retrofit of heat-integrated crude oil distillation systems. Ph.D. Thesis, University of Manchester Institute of Science and Technology , Manchester, U.K., 2002. (14) Dipama, J.; Teyssedou, A. Synthesis of heat exchanger networks using genetic algorithms. Appl. Therm. Eng. 2008, 28, 1763−1773. (15) Sivanandan, S. N.; Deepa, S. N. Introduction to Genetic Algorithms; Springer Verlag: Berlin, 2008. (16) Pariyani, A.; Gupta, A.; Ghosh, P. Design of heat exchanger networks using randomized algorithm. Comput. Chem. Eng. 2006, 30, 1046−1053. (17) Papoulias, S. A.; Grossmann, I. E. A structural optimization approach in process synthesis. II. Heat recovery networks. Comput. Chem. Eng. 1993, 7 (6), 707−721. (18) Lewin, D. R. A generalized method for HEN synthesis using stochastic optimizationII. The synthesis of cost-optimal networks. Comput. Chem. Eng. 1998, 22, 1387−1405. (19) Lin, B.; Miller, D. C. Solving heat exchanger network synthesis problems with Tabu search. Comput. Chem. Eng. 2004, 28 (8), 1451− 1464.

4. CONCLUSIONS In this article, an improved simultaneous optimization approach is proposed for heat exchanger network design. To improve the accuracy of this approach, varying stream heat capacities and the costs of mixers and splitters have been taken into consideration. In addition, a group of heuristic rules has been suggested for a user interaction stage to discard unreasonable matches in advance, which enables the design problem to be solved efficiently on ordinary desktop computers, and then an improved model for HEN design is constructed. A genetic algorithm was employed to select the best solution, which can avoid being trapped into local optimum. The whole procedure consists of two stages, namely, a user interaction stage and a mathematical stage. The key point of this approach is to exploit the interrelationships among streams. A case study confirms that this method provides a practical and effective procedure for HEN design.

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NOTE ADDED AFTER ASAP PUBLICATION This paper was published online April 23, 2012. Corrections were made to equations 18 and 20, and the corrected version was reposted April 27, 2012.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of China National offshore Oil Corporation project (No. 01018490125050029).

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REFERENCES

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