Simultaneous Heat Exchanger Network Synthesis Involving

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Simultaneous heat exchanger network synthesis involving nonisothermal mixing streams with temperature dependent heat capacity Hao Wu, Fangyou Yan, Wei Li, and Jinli Zhang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b01592 • Publication Date (Web): 31 Aug 2015 Downloaded from http://pubs.acs.org on September 7, 2015

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Industrial & Engineering Chemistry Research

Simultaneous heat exchanger network synthesis involving nonisothermal mixing streams with temperature dependent heat capacity

Hao Wua, Fangyou Yana, Wei Lia, Jinli Zhangb*

Key Laboratory for Green Chemical Technology MOE a and Key Laboratory of Systems Bioengineering MOE b, School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China

* Correspondence to: [email protected]

1

ACS Paragon Plus Environment


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Abstract: The heat exchanger network synthesis (HENS) is facing up with the dilemma of the substantial nonisothermal mixing and streams nonconstant thermal properties against the higher computational efficiency to solve mathematical programming models with large amount of variables and high nonlinearity. In this work, replacing the conventional binary variable in HENS model, a nonlinear approximation term is developed to indicate the existence of heat exchanger so as to reduce the amount of variables and constraint as well as enhance the model’s computational efficiency. With this approximation, a new HENS model was formulated considering simultaneously nonisothermal mixing and temperature-dependent heat capacity flow rate. All the variables of the model are stated with clear upper and lower bounds. A 10-stream (5 hot process streams and 5 cold process streams) benchmark HENS problem is solved by the developed model. The results show that the proposed model can generate a better cost-optimal heat exchanger network (HEN) with more accurate configuration in relatively short solution time compared with literatures.

Keywords: HENS, nonisothermal mixing, heat capacity flow rate, cost-optimal

2


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1 Introduction Energy consumption, as one of the global issues, has attracted lots of serious attentions to in recent years. Since the heat exchanger network (HEN) is a major energy consuming unit in chemical industry, the heat exchanger network synthesis (HENS) has been intensively studied in the field of chemical engineering as a systematic way to utilize energy efficiently and economically1. HENS methods can be mainly divided into two different categories. One is the pinch-analysis-based method and the other is mathematical-programming-based method2. The pinch analysis method has been playing an important role in solving industrial HENS problems3. While with a tremendous development of computing technology recently, the mathematical programming method gradually shows significant potential. Floudas and Ciric4 formulated a simultaneous mixed integer nonlinear programming (MINLP) model adopting the fixed heat recovery approach temperature (HRAT), which was the first to optimize heat exchanger location and network configuration simultaneously. Based on a stagewise superstructure (see Fig. 1), Yee and Grossmann5 formulated another simultaneous MINLP model, the SYNHEAT model, which did not rely on the presupposition of HRAT nor subnetwork partitioning. It is indicated that the SYNHEAT model can accomplish HENS optimally with no need of any manual interactions. Since the SYNHEAT model is a large, complex and non-convex MINLP model, generally it is difficult to find the globally optimal solution. Many efforts have been paid to optimize the solution procedure of the SYNHEAT model. Bergamini et al.6 3



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established a solution strategy based on the outer approximation which can generate the global optimal structure with operating conditions but also provide alternative structures. Bogataj and Kravanja7 proposed an alternative strategy of HENS global optimization based on a modified outer approximation/equality relaxation algorithm, together with an algorithm to identify high-quality local optimal solution. Huang and Karimi8 reported an efficient algorithm involving a repeating application of the outer approximation algorithm to solve MINLP problems efficiently. In particular, Faria et al.9 provided a brief review about HENS global optimization and put forward a bound contraction procedure to develop a new global optimization algorithm. It is worthwhile to note that the isothermal mixing assumption, i.e., all the substreams entering the same mixer are assumed at the same temperature5, makes it simplified to solve the original SYNHEAT model. However, such isothermal mixing assumption could probably omit some potential optimal HENS solutions.

Fig. 1. HEN superstructure5 Bjork and Westerlund10 formulated a nonisothermal mixing MINLP model using the same superstructure proposed by Yee and Grossmann5 through introducing some new variables (see Fig. 2). In Bjork and Westerlund10’s work, a HENS comparison was made by following and removing the isothermal mixing assumption. For both HENS methods, they developed a global optimization approach with piecewise approximation. Adopting a modified superstructure11, Huang et al.12 formulated a new MINLP model involving the nonisothermal mixing through introducing several 4


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bypass variables and improving temperature bounds and logical constraints. By employing the modified model, Huang et al.12 obtained the superior HEN for different scales of HENS problems and reported that the global optimization algorithm of Bjork and Westerlund10 failed to get the globally optimal solution for a relatively large problem (6 process streams, 2 hot streams and 4 cold streams). In the models of both Bjork and Westerlund10and Huang et al.12, constant heat-capacity flow rates were adopted because the nonconstant thermal properties can undoubtedly increase the nonlinearity of the nonisothermal mixing MINLP models, and consequently the extra variables and constraints could result in much more difficulty to solve. However, the substantial HENS sometimes possess the streams experiencing large temperature variations, in which the optimal solution could not be obtained under the assumption of constant thermal properties2, 13.

Fig. 2. New variables needed for nonisothermal mixing10. Smith et al.14 explored the multi-segment formulations for those streams with large temperature variations, considering the temperature-dependence of the stream thermal properties (such as heat capacity) in a HEN retrofitting problem. By breaking temperature range into several intervals in which the heat capacity is assumed constant, their methodology partially overcomes the disadvantage of constant heat capacity. Sreepathi and Rangaiah15 studied the same one retrofitting problem using elitist non-dominated sorting genetic algorithm with a continuous cubic polynomial heat capacity flow rate other than the stepwised constant heat capacity. Li et al.13 5



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proposed an improved simultaneous isothermal mixing HENS approach in which the streams’ heat capacity flow rate are calculated with a rigorous temperature dependent relationship. They adopted several heuristic rules to simplify the calculation procedure based on genetic algorithm because the nonconstant heat capacity flow rate could increase the computational difficulty. The nonisothermal mixing assumption can improve the HEN configuration and the nonconstant stream thermal properties can make the configuration closer to the reality. However, it requires higher computational efficiency to solve the HEN mathematical programming model in a reasonable time, taking into account simultaneously the nonisothermal mixing and nonconstant thermal properties. To solve such dilemma of HENS optimization problem, in this article, a nonlinear approximation term is developed to indicate the existence of heat exchanger, replacing the conventional binary variable. With this replacement, in Section 2, we established a nonisothermal mixing HENS model M1 and then extended M1 to the model M2 which simultaneously contains nonisothermal mixing and nonconstant heat capacity flow rate. And we stated all the variables’ logical bounds which may not be completely reported by literatures. In Section 3, six cases of HENS cases were solved using M1 with binary variables and M1 with the nonlinear approximations, respectively. The results show that our nonlinear approximation can enhances the model’s computational efficiency. And all the solutions yielded with M1 are comparable to or better than those reported in literatures. We used the benchmark 10SP1 problem, case 6, to study the impact of nonconstant heat capacity. The results 6


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show that the model M2 is superior to handle the nonisothermal mixing and nonconstant heat capacities simultaneously and can find a cost-optimal solution in reasonable time. To the best of our knowledge, no related model has been reported yet.

2 Mathematical formulation 2.1 M1: Nonisothermal mixing model The model M1 presented here is a stagewise HENS model with nonisothermal mixing streams. We adopted the superstructure proposed by Yee and Grossmann5 (see Fig. 1). In the superstructure, hot streams I exchange heat with cold streams J inside stages K. The adopted assumptions are shown below: 1) Utilities and process streams’ inlet and outlet temperatures are known as fixed value. 2) The heat capacity flow rate of every hot and cold stream is known as constant. 3) Utility is used only at the outlet of the last stage. 4) Matches between hot and hot (cold and cold) streams are forbidden. 5) All streams (substreams) entering the same mixer can be at different temperatures (nonisothermal mixing). Objective function TAC = ∑∑∑FCi , j i

j

k

qi , j ,k qi , j ,k + ε

+ ∑FCi ,cu i

qhu j qcui + ∑FC j , hu qcui + ε qhu j j +ε

 qi , j , k + ∑CCUqcui + ∑CHUqhu j + ∑∑∑CAijk   i j i j k  U i , j LMTDi , j , k  qcui + ∑CA i ,cu   i  Ui ,cu LMTDi ,cu

  

βi ,cu

 qhui + ∑CA j , hu   j  U j ,hu LMTD j ,hu

  

  

β i , j ,k

(1)

β j ,hu

The objective function is the general total annual cost (TAC) of the HEN. The 7



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optimization of HENS is to minimize TAC. In the TAC function, we used a nonlinear approximation term,

q , to denote whether or not a heat exchanger exists. ε is q +ε

set as 10-6 to make sure that the term equals 1 approximately when q > 0 and it equals 0 when q = 0. The function of the nonlinear term is the same with the conventional binary variable z in the HENS model. Introducing the nonlinear approximation, we can reduce the amount of variables, eliminate the binary variable type in the model and the conventional big M constraints, as well as transfer the MINLP model into NLP model. In the context we will prove that this nonlinear approximation can improve the solution procedure so as to enhance model’s computational efficiency. The logarithmic mean temperature difference (LMTD) approximation we used is proposed by Chen 16: 1

LMTDi , j ,k

(dthi , j ,k + dtci , j ,k )  3  =  dthi , j , k dtci , j ,k  , ∀i ∈ I , j ∈ J , k ∈ K 2  

LMTDi ,cu

( dtcui + dtcupi )  3  =  dtcui dtcupi  , ∀i ∈ I 2  

LMTD j ,hu

(dthu j + dthup j )  3  =  dthu j dthup j  , ∀j ∈ J 2  

(2)

1

(3)

1

(4)

The LMTDs don’t belong to variables because they are replaced by Eq. (2), (3), (4) directly in the model. We list Eq. (2), (3), (4) separately just for clarity. Parameters in the objective function can be calculated as below:

Ui , j =

1 , ∀i ∈ I , j ∈ J 1 1 + hi h j

(5)

8


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U i ,cu =

U j ,hu =

1 , ∀i ∈ I 1 1 + hi hcu

1 1 1 + h j hhu

(6)

, ∀j ∈ J

(7)

dtcupi = TOUTi − TIN cu , ∀i ∈ I

(8)

dthup j = TIN hu − TOUT j , ∀j ∈ J

(9)

Constraints Process stream overall heat balance:

Fi ( TINi − TOUTi ) = qcui + ∑∑qi , j ,k , ∀i ∈ I j

(10)

k

Fj ( TIN j − TOUT j ) = qhu j + ∑∑qi , j ,k , ∀j ∈ J i

(11)

k

Stagewise heat balance: Fi ( ti ,k − ti , k +1 ) = ∑qi , j ,k , ∀i ∈ I , k ∈ K

(12)

Fj ( t j ,k − t j , k +1 ) = ∑qi , j ,k , ∀j ∈ J , k ∈ K

(13)

j

i

Utility load balance: Fi ( ti ,NOK +1 − TOUTi ) = qi ,cu , ∀i ∈ I

(14)

Fj ( TOUT j − t j ,1 ) = q j ,hu , ∀j ∈ J

(15)

Temperature assignment:

ti ,1 = TINi , ∀i ∈ I

(16)

t j ,NOK +1 = TIN j , ∀j ∈ J

(17)

From above we can find out that the heat balance for each stage and the utility load balance can ensure the overall heat balance for each stream. Note that in some cases, overall heat balance constraints are very useful for reducing computational time to find the optimal solution. We compared the results with and without the overall 9



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heat balance constraints. And for some large cases, the results without these constraints are better (details can be found in the Supporting Information). Based on this, we decided to exclude Eq. (10), (11) out of the model. As shown in Fig. 2, the nonisothermal mixing model requires extra variables comparing with the isothermal mixing model. These variables’ constraints are shown below:

∑ xr

i, j ,k

= 1, ∀i ∈ I , k ∈ K

(18)

= 1, ∀j ∈ J , k ∈ K

(19)

j

∑ yr

i, j ,k

i

Relaxed heat balance for each heat exchanger10:

qi , j ,k ≤ xri , j ,k Fi (ti ,k − tihi , j ,k ), ∀i ∈ I , j ∈ J , k ∈ K

(20)

qi , j ,k ≤ yri , j ,k Fj (tici , j , k − t j ,k +1 ), ∀i ∈ I , j ∈ J , k ∈ K (21) Logical temperature constraints:

ti ,k ≥ ti ,k +1 , ∀i ∈ I , k ∈ K

(22)

t j ,k ≥ t j ,k +1 , ∀j ∈ J , k ∈ K

(23)

ti ,NOK +1 ≥ TOUTi , ∀i ∈ I

(24)

t j ,1 ≤ TOUTj , ∀j ∈ J

(25)

Hot end and cold end approach temperature constraints for heat exchanger:

 q EMAT ≤ dthi , j ,k ≤ ti ,k − tici , j ,k + Γ i , j ,k 1 − i , j ,k  q +ε i , j ,k 

  , ∀i ∈ I , j ∈ J , k ∈ K 

 qi , j ,k EMAT ≤ dtci , j ,k ≤ tihi , j ,k − t j ,k +1 + Γ i , j ,k  1 −  q +ε i , j ,k 

  , ∀i ∈ I , j ∈ J , k ∈ K 

Hot end approach temperature constraint for cooler:

10


(26)

(27)

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 qcui  EMAT ≤ dtcui ≤ ti ,NOK +1 − TOUTcu + Γ i ,cu 1 −  , ∀i ∈ I  qcui + ε  (28) Cold end approach temperature constraint for heater:  qhu j EMAT ≤ dthu j ≤ TOUThu − t j ,1 + Γ j , hu  1 −  qhu + ε j 

  , ∀j ∈ J 

(29) The parameter Γ is calculated as below: if TIN i ≥ TIN j +EMAT  TIN i − TIN j  abs (TIN i − TIN j ) + EMAT otherwise

Γ i , j ,k = 

(30)

if TIN i ≥ TIN cu +EMAT  TIN i − TIN cu  abs (TIN i − TIN cu ) + EMAT otherwise

Γ i ,cu = 

(31) if TIN hu ≥ TIN j +EMAT  TIN hu − TIN j  abs (TIN hu − TIN j ) + EMAT otherwise

Γ j ,hu =  (32)

In order to avoid infinite heat exchanger areas, a small positive bound, EMAT, is introduced for the approach temperatures. The default value of EMAT is set as 0.1 K in this work. Eq. (1) ~ (9) and (12) ~ (32) construct the nonisothermal mixing model

M1, through substituting the binary variables in these equations by the nonlinear term q . q+ε

2.2 M2: Nonisothermal mixing model involving streams with nonconstant heat capacity Previously, Li et al.13 demonstrated that different ways to handle the stream’ heat capacity could generate different HENS results. It is necessary to formulate a model 11



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in which all the process streams have realistic or temperature dependent heat capacities. Taking into account the nonconstant heat capacity of the model, Eq. (33) is adopted to identify the correlation between heat capacity (cp) and temperature (T)17.

cp = a0 + a1T + a2T 2 + a3T 3 (33) The heat capacity flow rate is calculated by Eq. (34) where f is the stream’s flow rate. F = f ( a0 + a1T + a2T 2 + a3T 3 )

(34) Due to the nonconstant process stream heat capacity flow rate, the heat load of a heat exchanger or a network stage should be calculated by an integral:

Q=∫

TOUT

TIN

f (a0 + a1T + a2T 2 + a3T 3 )dT

(35) Thus we can formulate M2, an extended version of M1. The equations in M2 and

M1 are all the same except that in M2, Eq. (12) ~ (15), (20) and (21) which use constant heat capacity flow rates should be rewritten as below.

∑q

=∫

i , j ,k

ti ,k

fi (ai ,0 + ai ,1T + ai ,2T 2 + ai ,3T 3 )dT , ∀i ∈ I , k ∈ K

ti ,k +1

j

(36)

∑q

=∫

i, j ,k

t j ,k

t j ,k +1

i

f j (a j ,0 + a j ,1T + a j ,2T 2 + a j ,3T 3 ) dT , ∀j ∈ J , k ∈ K

(37)

qcu ,i = ∫

ti ,NOK +1

TOUTi

f i (ai ,0 + ai ,1T + ai ,2T 2 + ai ,3T 3 ) dT , ∀i ∈ I

(38) 12


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qhu , j = ∫

TOUT j

f j (a j ,0 + a j ,1T + a j ,2T 2 + a j ,3T 3 ) dT , ∀j ∈ J

t j ,1

(39) ti , k

qi , j , k ≤ xri , j ,k ∫

tihi , j ,k

f i (ai ,0 + ai ,1T + ai ,2T 2 + ai ,3T 3 ) dT , ∀i ∈ I , j ∈ J , k ∈ K

(40) qi , j , k ≤ yri , j ,k ∫

tici , j ,k

t j ,k +1

f j (a j ,0 + a j ,1T + a j ,2T 2 + a j ,3T 3 ) dT , ∀i ∈ I , j ∈ J , k ∈ K

(41)

2.3 Variable bounds When a model is formulated, all the involved variables are suggested to be defined with lower bounds and upper bounds. These variable bounds are very helpful to improve the model’s precision and reduce the computational difficulty. All the involved variable’s logical bounds we used are listed below: 0 ≤ qi , j ,k ≤ min ( Qi , Q j ) , ∀i ∈ I , j ∈ J , k ∈ K

(42) 0 ≤ qcui ≤ Qi , ∀i ∈ I (43) 0 ≤ qhu j ≤ Q j , ∀j ∈ J

(44)

0 ≤ xri , j ,k ≤ 1, ∀i ∈ I , j ∈ J , k ∈ K

(45)

0 ≤ yri , j ,k ≤ 1, ∀i ∈ I , j ∈ J , k ∈ K (46)

TOUTi ≤ ti ,k ≤ TINi , ∀i ∈ I , k ∈ K ∪ { NOK + 1} TIN j ≤ t j ,k ≤ TOUTj , ∀j ∈ J , k ∈ K ∪ { NOK + 1} (48)

13


(47)


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 TIN j + EMAT ≤ tihi , j , k ≤ TIN i  TOUTi ≤ tihi , j ,k ≤ TIN i

if TIN i > TIN j + EMAT

 TIN j ≤ tici , j ,k ≤ TIN i − EMAT  TIN j ≤ tici , j , k ≤ TOUT j

if TIN i > TIN j + EMAT

otherwise

otherwise

EMAT ≤ dthi , j , k ≤ 2Γ i , j , k , ∀i ∈ I , j ∈ J , k ∈ K

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(49)

(50) (51)

EMAT ≤ dtci , j ,k ≤ 2Γ i , j ,k , ∀i ∈ I , j ∈ J , k ∈ K (52)

EMAT ≤ dtc j ,hu ≤ 2Γ j ,hu , ∀j ∈ J (53)

EMAT ≤ dthi ,cu ≤ 2Γ i ,cu , ∀i ∈ I

(54)

2.4 Solution strategy The above two models are formulated in the General Algebraic Modeling System (GAMS). Since the nonisothermal mixing brings various extra variables into the HENS model, manually find a good initial feasible solution seems to be hard. Therefore we adopted the BARON as the global solver instead of DICOPT because BARON does not need any initial value to solve the model. We set the default maximum solution time as 1 hour since BARON is generally slow. We used GAMS v24.2.2 with BARON as the global solver to solve the mathematical models. The computing platform is a laptop with 2.5 GHz Core® I5-4200M CPU and 4 GB of RAM.

3 Case Study All the six cases’ problem data can be found in the Supporting Information. Case 1

14


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This case from Lorenz T. Biegler et al.18 has 3 process streams (1 hot stream and 2 cold streams). Case 1 is solved to validate the model. Fig. 3 shows the generated HEN structure. It has three exchangers and a TAC of $76,444, which is similar with the value of $76,445 reported by Lorenz T. Biegler et al.18 but is higher than Huang et al.12’s best result $76,327. The difference is attributed to the different LMTD approximation we adopted. That’s the reason Huang et al.12 studied the LMTD approximations’ impact to HENS.

Fig. 3. Optimal HEN configuration for case 1. Case 2 This case is from Bjork and Westerlund10. It contains 3 hot process streams and 2 cold process streams. As Bjork and Westerlund10, this case is studied as a middle scale example to show the impact of nonisothermal mixing. As displayed in Fig. 4, the optimal HEN shows a TAC of $95,723, which is lower than the value of $96,001 reported by Bjork and Westerlund10. The reason of the difference is the same as that of case 1.

Fig. 4. Optimal HEN configuration for case 2. Case 3 Like case 2, this case (6 process streams) is also from Bjork and Westerlund10. It’s the largest nonisothermal mixing problem reported by Bjork and Westerlund10, with FCi,j,k =$8000, which was selected to analyze the effect of our nonlinear approximation in the objective function on the solution. Fig. 5 shows the optimal 15



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HEN, with the TAC of $128,389 which is lower than $139,083 reported by Bjork and Westerlund10 adopting their nonisothermal mixing model.

Fig. 5. Optimal HEN configuration for case 3. Case 4 This case comes from Yee and Grossmann5. Its main purpose is to study the HENS models based on isothermal mixing and non-isothermal mixing assumption. Because this 6-stream case has only one cold stream, so there is a very high possibility to generate an HEN with splitting and mixing streams. The HEN configuration generated by M1 is shown in Fig. 6, with the TAC of $571,841, which is lower than $572,476 reported by Khorasany and Fesanghary19.

Fig. 6. Optimal HEN configuration for case 4. Case 5 This case is originally from Ahmad20. It is a HENS benchmark which has been studied by Ravagnani et al.21, Yerramsetty and Murty22 and Khorasany and Fesanghary19. In the original version, the exchanger fixed cost FCi,j,k=0. Huang et al.12 modified the case by proposing FCi,j,k =$8000. We solved the case following this modification. Fig. 7 shows the HEN configuration generated by M1, with the best TAC of $5,722,602, which is less than $5,810,558 reported by Huang et al.12 with the same LMTD approximation.

Fig. 7. Optimal HEN configuration for case 5. As listed in Table 1, we compared our results of case 1-5 with those in the 16


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literature. Huang et al.12 studied all the five cases using an improved nonisothermal mixing HENS model with several binary bypass variables and the global solver of BARON as well as the same LMTD expression proposed by Chen16. It is clear that the TACs of cases 1-4 calculated in this work are the same as those of Huang et al.12, while the TAC of case 5 is lower here, suggesting the precision and superior calculation efficiency of our established model M1. Moreover, Bjork and Westerlund10 studied case 1-3 with the LMTD proposed by Paterson23 using DICOPT solver. Their TAC of case 1 is better than ours, but they didn’t report their HEN configuration. Khorasany and Fesanghary19 reported their best TAC of case 4 as $572,476 which is slightly higher than ours. Their algorithm is not deterministic so they reported an average solution time of 30 runs. Huang and Karimi8 reported the same configuration as ours for case 1 but since the LMTD is different, their TAC is lower. By allowing cross flow in the superstructure, Huang and Karimi8 yielded better cost-optimal solutions for case 2 and 3. But for case 5, our solution is still superior.

Table 1. Comparisons of the TAC and the CPU time for cases 1~5. Ref.

Case 1

Case 2

Case 3

Case 4

Case 5

TAC($)

76,444

95,723

128,389

571,841

5,722,602

Time(s)

(1.37)

(17)

(55)

(812)

(2991)

TAC($)

76,444

95,723

128,389

571,841

5,810,558

Time(s)

(5)

(40)

(3816)

(928)

(8136)

Bjork and Westerlund10

TAC($)

76,330

96,001

139,083

Time(s)

(938)

(96)

(193)

-

-

Khorasany and Fesanghary19

TAC($)

-

-

-

Huang and Karimi8

TAC($)

76,327

94,742

123,357

Time(s)

(8)

(26)

(109)

This work Huang et al.12

Time(s)

*: average time of 30 runs. 17


572,476 (903*) -

5,733,679 (2387)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Case 6 This is a published case with 10 process streams (5 hot streams and 5 cold streams). It has been reported by Li et al.13, Yerramsetty and Murty22, Papoulias and Grossmann24, Lewin et al.25, Lin and Miller26, Pariyani et al.27, Pettersson28, Gupta and Ghosh29, and He and Cui30. Among those literatures, only Li et al.13 built a polynomial correlation of Fi, Fj and T by operating regression analysis of ASPEN PLUS simulation results.

Fig. 8. Optimal HEN configuration for case 6. The final optimal HEN configuration generated by M2 including stream temperatures before and after mixers, inlet and outlet temperatures of stages, and exchanger heat loads shows in Fig. 8, the total annual cost is $42,991.6. This network consists 8 heat exchangers within 2 stages and 2 coolers for the hot streams H4 and H5. As shown in Fig. 8, there are four mixers in the HEN and the configuration denotes all the four mixers are nonisothermal. Traditional isothermal mixing assumption will omit this configuration because the exchanger temperature approach calculated by nonisothermal mixing is different, for example, the hot end approach temperature for HE4,5,1 is 32 oC in the presented HEN configuration, while if isothermal mixing assumption is adopted, the hot end approach temperature will be calculated as only 5 oC. It is clear that the model presented here is superior to get the cost-optimal solution, whereas the more nonisothermal mixers could result in the network complexity and operating difficulty. 18


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Table 2. Comparisons of the results for case 6 using different models. QH QC units Area Time (s) (kW) (kW) (m2) 0 1879 10 0 1879 10 78 0 1879 10 28,207.1 0 1879 10 10,012.3 0 1879 10 0 1879 10 248.8 0 1879 10 0 1879 10 6842 0 1879 10 0 1879 10 259.4 0 1879 10 233.6 730 0 1879 10 223.7 2970 a nonconstant heat capacity is assumed

TAC ($/yr) 43,439 43,452 43,329 43,439 43,538 43,331 43,342 43,602a 43,392 43,570 42,962 42,992a

Ref. Papoulias and Grossmann24 Lewin et al.25 Lin and Miller26 Pariyani et al.27 Yerramsetty and Murty22 Pettersson28 Gupta and Ghosh29 Li et al.13 He and Cui30 Escobar and Trierweiler31 M1 M2

Table 2 lists the results of case 6 calculated using different models. The HEN configuration generated by M1 can be found in the Supporting Information. All the literature report the same results as ours including 0 hot utility, 1879 kW cold utility and 10 heat exchanger units, whereas the hear exchange area and the TAC values calculated by our models are lower than others. It is clear that the TAC value using

M1 (with constant heat capacity) is better than that using M2, suggesting that the model M2 (with nonconstant heat capacity) is not necessary to provide the better TAC than M1 but the generated network configuration using M2 is closer to the reality (temperature dependent heat capacity). It is worthwhile to note that the M2’s solution time is longer than M1, indicating that introducing the nonlinearity into the model can make the solution procedure a little more difficult although the nonlinear approximation can reduce the amount of variables and eliminate the binary variable in the model. Table 3 compares the results calculated using our nonlinear approximation and 19



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Page 20 of 33

those with conventional binary variable for these six cases. It is indicated that adopting the binary variable cases 1~5 can only find a local optimal solution while the case 6 yields no feasible solution , given the maximum CPU time of 3600 s. Thus, our nonlinear approximation indeed enhances the model’s computational efficiency, other than making the solution procedure more difficult.

Table 3. Comparing results using the nonlinear approximation with those using binary variables. Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

With nonlinear

TAC($/year)

76,444

95,723

128,389

571,841

5,722,602

42,992

approximation

CPU time(s)

1.37

17

55

812

2991

2970

TAC($/year)

77,074

121,045

146,440

612,089

7,343,427

-

CPU time(s)

126

1594

1880

2130

1184

3600

With binary variables

Case 1-5 are solved with M1, Case 6 is solved with M2.

4 Conclusion We described a detailed mathematical approach of HENS. The nonisothermal mixing assumption is adopted to explore better cost-optimal HEN configuration, and considering the nonconstant heat capacities for all the process streams makes the configuration closer to the reality. Adopting our nonlinear approximation method, the formulated model can take nonisothermal mixing and nonconstant heat capacity into consideration simultaneously without any computational difficulty. A case study shows that we can get a better cost-optimal HEN with more accurate configuration by solving the proposed model. To some extent, the HEN structure is relative complicated because the nonisothermal mixing assumption always results in a HEN with splitting streams. Further work can focus on this point to make the HEN configuration more practical. 20


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Acknowledgment This work was supported by the Special Funds for Major State Research Program of China (2012CB720300), Program for Changjiang Scholars and Innovative Research Team in University (IRT1161).

Nomenclature Abbreviations HE

heat exchanger

HEN

heat exchanger network

HENS heat exchanger network synthesis HRAT heat recovery approach temperature LMTD logarithmic mean temperature difference MINLP mixed-integer nonlinear programming NOK

number of K

TAC

total annual cost

Set I

set of hot streams

J

set of cold streams+

K

set of stages

K ∪ { NOK + 1} set of temperature locations Subscripts i

hot process stream

j

cold process stream 21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

k

stage/temperature location

cu cold utility hu hot utility Parameters EMAT positive bound for heat exchanger approach temperature TAC

objective total annual cost

a

coefficient related with stream heat capacity

ε

small number

f

flow rate

F

heat capacity flow rate

FC

fixed charge of heat exchanger

CCU

cold utility cost

CHU

hot utility cost

CA

area cost coefficient of heat exchanger

U

overall heat transfer coefficient

TIN

inlet temperature

TOUT outlet temperature hi

film heat transfer coefficient

Γ

upper bound of approach temperature

dtcup

cold end temperature approach

dthup

hot end temperature approach

Q

heat exchange duty 22


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β

exponent of area cost for heat exchanger

Non-negative variables t

temperature

q

heat load of heat exchanger

dth

hot end approach temperature of heat exchanger

dtc

cold end approach temperature of heat exchanger

dtcu

hot end approach temperature for cooler

dthu

cold end approach temperature for heater

xr

flow rate fraction of hot stream in heat exchanger

yr

flow rate fraction of cold stream in heat exchanger

tihi,j,k

hot stream’s cold end temperature of heat exchanger

tici,j,k

cold stream’s hot end temperature of heat exchanger

23



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References 1.

Furman, K. C.; Sahinidis, N. V. A Critical Review and Annotated Bibliography for Heat

Exchanger Network Synthesis in the 20th Century. Ind. Eng. Chem. Res. 2002, 41, (10), 2335-2370. 2.

Sreepathi, B. K.; Rangaiah, G. P. Review of Heat Exchanger Network Retrofitting Methodologies

and Their Applications. Ind. Eng. Chem. Res. 2014, 53, (28), 11205-11220. 3.

Linnhoff, B.; Hindmarsh, E. The Pinch Design Method for Heat Exchanger Networks. Chem. Eng.

Sci. 1983, 38, (5), 745-763. 4.

Floudas, C. A.; Ciric, A. R. Strategies for Overcoming Uncertainties in Heat Exchanger Network

Synthesis. Comput. Chem. Eng. 1989, 13, (10), 1133-1152. 5.

Yee, T. F.; Grossmann, I. E. Simultaneous Optimization Models for Heat Integration—Ii. Heat

Exchanger Network Synthesis. Comput. Chem. Eng. 1990, 14, (10), 1165-1184. 6.

Bergamini, M. L.; Scenna, N. J.; Aguirre, P. A. Global Optimal Structures of Heat Exchanger

Networks by Piecewise Relaxation. Ind. Eng. Chem. Res. 2007, 46, (6), 1752-1763. 7.

Bogataj, M.; Kravanja, Z. An Alternative Strategy for Global Optimization of Heat Exchanger

Networks. Appl. Therm. Eng. 2012, 43, 75-90. 8.

Huang, K. f.; Karimi, I. A. Efficient Algorithm for Simultaneous Synthesis of Heat Exchanger

Networks. Chem. Eng. Sci. 2014, 105, 53-68. 9.

Faria, D. C.; Kim, S. Y.; Bagajewicz, M. J. Global Optimization of the Stage-Wise Superstructure

Model for Heat Exchanger Networks. Ind. Eng. Chem. Res. 2015, 54, (5), 1595-1604. 10. Bjork, K. M.; Westerlund, T. Global Optimization of Heat Exchanger Network Synthesis Problems with and without the Isothermal Mixing Assumption. Comput. Chem. Eng. 2002, 26, (11), 1581-1593. 11. Hasan, M. M. F.; Jayaraman, G.; Karimi, I. A.; etc. Synthesis of Heat Exchanger Networks with Nonisothermal Phase Changes. AlChE J. 2010, 56, (4), 930-945. 12. Huang, K. F.; Al-mutairi, E. M.; Karimi, I. A. Heat Exchanger Network Synthesis Using a Stagewise Superstructure with Non-Isothermal Mixing. Chem. Eng. Sci. 2012, 73, 30-43. 13. Li, G.; Luo, Y.; Xia, Y.; etc. Improvement on the Simultaneous Optimization Approach for Heat Exchanger Network Synthesis. Ind. Eng. Chem. Res. 2012, 51, (18), 6455-6460. 14. Smith, R.; Jobson, M.; Chen, L. Recent Development in the Retrofit of Heat Exchanger Networks. Appl. Therm. Eng. 2010, 30, (16), 2281-2289. 15. Sreepathi, B. K.; Rangaiah, G. P. Retrofitting of Heat Exchanger Networks Involving Streams with Variable Heat Capacity: Application of Single and Multi-Objective Optimization. Appl. Therm. Eng. 2015, 75, (0), 677-684. 16. Chen, J. J. J. Comments on Improvements on a Replacement for the Logarithmic Mean. Chem. Eng. Sci. 1987, 42, (10), 2488-2489. 17. Sandle, S. I. Chemical and Engineering Thermodynamics. 3rd ed.; Wiley: New York, 1999. 18. Lorenz T. Biegler; Ignacio E. Grossmann; Westerberg, A. W. Systematic Methods of Chemical Process Design. Prentice Hall PTR: 1997. 19. Khorasany, R. M.; Fesanghary, M. A Novel Approach for Synthesis of Cost-Optimal Heat Exchanger Networks. Comput. Chem. Eng. 2009, 33, (8), 1363-1370. 20. Ahmad, S. Heat Exchanger Networks: Cost Trade-Offs in Energy and Capital. UMIST, Manchester,UK, 1985. 21. Ravagnani, M. A. S. S.; Silva, A. P.; Arroyo, P. A.; etc. Heat Exchanger Network Synthesis and 24


Page 24 of 33

Page 25 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Optimisation Using Genetic Algorithm. Appl. Therm. Eng. 2005, 25, (7), 1003-1017. 22. Yerramsetty, K. M.; Murty, C. V. S. Synthesis of Cost-Optimal Heat Exchanger Networks Using Differential Evolution. Comput. Chem. Eng. 2008, 32, (8), 1861-1876. 23. Paterson, W. R. A Replacement for the Logarithmic Mean. Chem. Eng. Sci. 1984, 39, (11), 1635-1636. 24. Papoulias, S. A.; Grossmann, I. E. A Structural Optimization Approach in Process Synthesis—Ii: Heat Recovery Networks. Comput. Chem. Eng. 1983, 7, (6), 707-721. 25. Lewin, D. R.; Wang, H.; Shalev, O. A Generalized Method for Hen Synthesis Using Stochastic Optimization - I. General Framework and Mer Optimal Synthesis. Comput. Chem. Eng. 1998, 22, (10), 1503-1513. 26. Lin, B.; Miller, D. C. Solving Heat Exchanger Network Synthesis Problems with Tabu Search. Comput. Chem. Eng. 2004, 28, (8), 1451-1464. 27. Pariyani, A.; Gupta, A.; Ghosh, P. Design of Heat Exchanger Networks Using Randomized Algorithm. Comput. Chem. Eng. 2006, 30, (6-7), 1046-1053. 28. Pettersson, F. Heat Exchanger Network Design Using Geometric Mean Temperature Difference. Comput. Chem. Eng. 2008, 32, (8), 1726-1734. 29. Gupta, A.; Ghosh, P. A Randomized Algorithm for the Efficient Synthesis of Heat Exchanger Networks. Comput. Chem. Eng. 2010, 34, (10), 1632-1639. 30. He, Q. L.; Cui, G. M. A Principle of Stream Arrangement Based on Uniformity Factor for Heat Exchanger Networks Synthesis. Appl. Therm. Eng. 2013, 61, (2), 93-100. 31. Escobar, M.; Trierweiler, J. O. Optimal Heat Exchanger Network Synthesis: A Case Study Comparison. Appl. Therm. Eng. 2013, 51, (1-2), 801-826.

25



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Figures

Fig. 1

26


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Fig. 2

27



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Fig. 3

28


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Fig. 4

29



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Fig. 5

30


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Fig. 6

31



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Fig. 7

32


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Fig. 8

33


Simultaneous Heat Exchanger Network Synthesis Involving

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