Optimal design of heat exchanger networks

CHAPTER 6

Optimal design of heat exchanger networks Wilfried Roetzela, Xing Luob, Dezhen Chenc a

Institute of Thermodynamics, Helmut Schmidt University/University of the Federal Armed Forces Hamburg, Hamburg, Germany Institute of Thermodynamics, Gottfried Wilhelm Leibniz University Hannover, Hannover, Germany c Institute of Thermal Energy and Environmental Engineering, Tongji University, Shanghai, China b

In many industrial processes, heat exchanger networks are used to transfer heat among more than two process streams, in which the cold streams are heated by the hot streams that need to be cooled and vice versa. In this way, a large amount of heat energy can be recovered from the process streams, and therefore, the heating and cooling loads from external sources (hot and cold utilities) can be dramatically reduced. However, the use of heat exchangers for heat recovery also increases the investment costs. Therefore, a balance between the utility costs and investment costs should be established. The optimal design of a heat exchanger network is to configure a heat recovery system or retrofit an existing network capable of performing the prescribed tasks at the minimum total annual costs that are mainly determined by the utility costs and investment costs (Masso and Rudd, 1969). Because of its structural characteristics, it is also named as synthesis of heat exchanger networks. The optimal design also deals with the optimal retrofit design of an existing heat exchanger network by rematching the process streams; changing the heat exchanger area; and adding or removing some heat exchangers, heaters, and coolers, so that the sum of the annual utility costs and retrofitting costs reaches the minimum. The available optimization design and synthesis methods can be classified into three categories: (1) thermodynamic analysis methods with pinch technology (Linnhoff and Flower, 1978a; Linnhoff et al., 1979); (2) mathematical programming methods (Grossmann and Sargent, 1978); and (3) stochastic or heuristic algorithms such as genetic algorithm (Lewin, 1998), simulated annealing algorithm (Dolan et al., 1989) and particle swarm Design and Operation of Heat Exchangers and their Networks https://doi.org/10.1016/B978-0-12-817894-2.00006-6

© 2020 Elsevier Inc. All rights reserved.

231

232

Design and operation of heat exchangers and their networks

optimization algorithm (Silva et al., 2010), and knowledge-based system (Expert System) (Chen et al., 1989; Souto et al., 1992). Till now, researches are still progressing along these three lines with the most attention to the latter two methods. To solve real-life industrial problems, the engineer should take advantage of all these disciplines. There are three areas of heat exchanger network synthesis: targeting, synthesis, and optimization. Targets include energy consumption (utilities), heat transfer area, number of heat exchange units, and finally total annual cost. The targets can be served as a motivation or to give the designer confidence that a network is close to “optimal.” Synthesis methods include the matching of hot and cold streams and the sequencing of the resulting heat exchangers. Optimization involves both topological and parameter improvements that reduce the total annual cost. According to whether the three elements consisting the total cost, namely, utilities, area and unit number, are considered simultaneous or separately, the available synthesis methods can be reclassified into two categories: targeting sequential methods and simultaneous synthesis methods. The methods in the first group progressively cut down the problem feasible region by successively imposing a series of design targets arranged by their decreasing impact on the total annual cost of the network. Usually, the top goal is the least utility usage to be achieved through a minimum number of units (the second-level target) at the lowest capital investment (the bottomlevel target). Though one cannot guarantee that a sequential method ends up with the network featuring the lowest total annual cost, it often provides a very good network design. Pinch design techniques are the typical representative in this group of sequential methods. The methods in the second group are aimed at finding the optimal heat exchanger network in a single step. These methods are no longer based on the assumption that the total annual cost is dominated by the utility requirements, and all of methods in the second group belong to the mathematical programming area and use a mixed-integer nonlinear programming (MINLP) problem formulation to seek the heat exchanger network featuring the least total cost at once. The design objective includes a quantitative part (cost of heat exchange equipment and external utilities) and a qualitative part (safety, operability, flexibility, and controllability). The quantitative part is the main topic of this chapter, and the qualitative part will be discussed in Chapter 9.

Optimal design of heat exchanger networks

233

6.1 Mathematical model and its general solution for rating heat exchanger networks Thermal analysis of a heat exchanger network is a basis for its optimal design, synthesis, regulation, and online control. Usually, the calculation is difficult because the inlet fluid temperatures of some heat exchangers in the network are unknown. For a heat exchanger network with a simple sequential arrangement of heat exchangers, the exit stream temperatures of the network can be obtained by calculating the outlet fluid temperatures of each heat exchanger sequentially. However, a practical network might have loops and branches; therefore, the unknown inlet fluid temperatures of some heat exchangers have to be assumed. To avoid an arduous iterative calculation, we introduce an explicit analytical solution for thermal calculation of heat exchanger networks (Roetzel and Luo, 2005; Chen et al., 2007). Consider a heat exchanger network having NE heat exchangers, NM mixers, N 0 stream entrances, and N 00 stream exits. In each heat exchanger, there are two fluid channels for hot and cold streams, respectively. A mixer is used to express a node at which two or more streams are mixed together and splitted again. One mixer is regarded as one channel. Therefore, the total number of channels N ¼ 2NE + NM. The channel indexes are related to the exchanger indexes, that is, the index of the hot stream in the jth exchanger is 2j  1 and that of the cold stream is 2j. The index of the mth mixer is 2NE + m. The indexes of the network entrances and network exits can be arbitrarily labeled. The outlet stream temperatures of the jth heat exchanger can be expressed as  00     tE, h, j vhh vhc tE0 , h, j ðj ¼ 1, 2, …, NE Þ (6.1) ¼ tE00, c, j vch vcc tE0 , c, j or in the matrix form T00E, j ¼ Vj T0E, j ðj ¼ 1, 2, …, NE Þ

(6.2)

0 00 in which TE, j and TE, j are the inlet and outlet temperature vectors of the jth heat exchanger, respectively. The coefficient matrix Vj can be calculated with Eq. (6.3):     v εj 1  εj v (6.3) Vj ¼ hh, j hc, j ¼ vch, j vcc, j Rj εj 1  Rj εj

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Design and operation of heat exchangers and their networks

where εj is the dimensionless temperature change of the hot fluid εj ¼

tE0 , h, j  tE00, h, j tE0 , h, j  tE0 , c, j

1  eNTUj ð1Rj Þ ∗ 1  R eNTUj ð1Rj Þ ∗

¼

(6.4)

j

Rj is the ratio of thermal capacity rates Rj ¼ C_ E, h, j =C_ E, c, j

(6.5)

and NTU∗j is the number of transfer units as a counterflow heat exchanger NTU∗j ¼ ðFkAÞE, j =C_ E, h, j Following special cases should be considered in the calculation coefficient matrix Vj:   0 1 ∗ NTUj ! ∞ and Rj  1 : V ¼ Rj 1  Rj   1  1=Rj 1=Rj ∗ NTUj ! ∞ and Rj > 1 : V ¼ 1 0 2 3 NTU∗j 1 6 1 + NTU∗ 1 + NTU∗ 7 6 j j 7 Rj ¼ 1 : V ¼ 6 7 4 NTU∗j 5 1 ∗ ∗ 1 + NTUj 1 + NTUj   NTU∗j NTU∗j 1  e e R j ¼ 0 : Vj ¼ 0 1   1 0 ∗ ∗ Rj ! ∞ : V ¼ 1  eNTUc, j eNTUc, j

(6.6) of the

(6.7) (6.8)

(6.9)

(6.10) (6.11)

where NTU∗c, j ¼ ðFkAÞE, j =C_ E, c, j

(6.12)

For a counterflow heat exchanger, the correction factor of mean temperature F ¼ 1. The equations of F for some typical types of heat exchangers can be found in Chapter 3. Extending Eq. (6.2) to the whole network yields the relation between the inlet and outlet temperature vectors as follows: T00E ¼ VT0E

(6.13)

Optimal design of heat exchanger networks

235

in which  T T0E ¼ tE0 , 1 tE0 , 2 ⋯ tE0 , N  T 0 0 0 ¼ th0 , 1 tc0 , 1 th0 , 2 tc0 , 2 ⋯ th0 , NE tc0 , NE tM (6.14) , 1 tM, 2 ⋯ tM, NM  T T00E ¼ tE00, 1 tE00, 2 ⋯ tE00, N  T 00 00 00 ¼ th00, 1 tc00, 1 th00, 2 tc00, 2 ⋯ th00, NE tc00, NE tM (6.15) , 1 tM, 2 ⋯ tM, NM 3 2 V1 0 7 6 ⋱ 7 6 7 6 V N E 7 (6.16) VN N ¼ 6 7 6 1 7 6 4 ⋱ 5 0 1 To illustrate the interconnections among the heat exchangers, we use the following four matching matrices (The first three have been introduced in Section 3.6). Interconnection matrix G: N  N matrix whose elements gij are defined as the ratio of the thermal capacity rate flowing from channel j into channel i to that flowing through channel i. Entrance matching matrix G0 : N  N 0 matrix whose elements g0 ik are defined as the ratio of the thermal capacity rate flowing from the entrance k to channel i to that flowing through channel i. Exit matching matrix G00 : N 00  N matrix whose elements g00 li are defined as the ratio of the thermal capacity rate flowing from channel i to the exit l to that flowing out of exit l. Bypass matrix G000 : N 00  N 0 matrix whose elements g000 lk are defined as the ratio of the thermal capacity rate flowing from entrance k to exit l to that flowing out of exit l. We can write the energy balances at the inlets of N channels and at the network exits of N 00 streams with these matrices as follows: tE0 , i

¼

N0 X

gik0 tk0 +

k¼1

tl00

¼

N0 X k¼1

glk000 tk0 +

N X

gij tE00, j ði ¼ 1, 2, …, N Þ

(6.17)

gli00 tE00, i ðl ¼ 1, 2, …, N 00 Þ

(6.18)

j¼1 N X i¼1

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Design and operation of heat exchangers and their networks

or in a matrix form: T0E ¼ G0 T0 + GT00E

(6.19)

T00 ¼ G000 T0 + G00 T00 E

(6.20)

0 00 and T00 ¼ ½ t100 t200 ⋯ tN are the entrance where T0 ¼ ½ t10 t20 ⋯ tN 0 00  and exit stream temperatures of the network, respectively. As has been shown in Section 3.6, substituting Eq. (6.13) into Eqs. (6.19), (6.20), we can explicitly express the inlet and outlet fluid temperatures of individual heat exchangers and the exit stream temperatures of the network with Eqs. (6.21), (6.13), (6.22), respectively. T

T

T0E ¼ ðI  GVÞ1 G0 T0   T00 ¼ G000 T0 + G00 T00E ¼ G000 + G00 VðI  GVÞ1 G0 T0

(6.21) (6.22)

Because Eqs. (6.21), (6.22) contain the calculation of an N  N inverse matrix, for a large heat exchanger network, more computing time might be required. As an alternative, the “upwind” iterative calculation method is recommended. The convergence of the iteration is ensured by alternately recalculating the temperatures of the hot/cold process streams with fixed cold/hot stream temperatures. (1) At the beginning, set all unknown inlet and outlet temperatures of the hot and cold process streams equal to their supply temperatures of the network. (2) Starting from the network entrances of the hot process streams and along the stream flow direction, calculate the outlet temperature of hot stream in each heat exchanger according to its known or assumed inlet temperature of the hot and cold streams but do not calculate the outlet temperature of the cold stream. (3) Starting from the network entrances of the cold process streams and along the stream flow direction, calculate the outlet temperature of cold stream in each heat exchanger according to its known or calculated inlet temperature of the hot and cold streams but do not calculate the outlet temperature of the hot stream. (4) Repeat Steps (2) and (3) to correct the outlet temperatures of hot and cold streams, respectively, until the given accuracy is achieved.

Optimal design of heat exchanger networks

237

Example 6.1 Rating a heat exchanger network. This example is taken from Toffolo (2009, Table 6.22). The revised heat exchanger network is shown in Fig. 6.1, in which the supply and target stream temperatures are expressed in bold at the left and right ends of the network and utilities, and the thermal capacity rates are given in brackets. It has six process heat exchangers, one mixer, one heater, and one cooler; therefore, there are 13 channels (excluding the heater and cooler). All units are counterflow heat exchangers and have the overall heat transfer coefficient of 1 kW/m2K. The hot and cold utility costs are 140 $/kW/yr and 10 $/kW/yr, respectively. The annual investment costs of the units are calculated with CE ¼ 1200A0.6$/yr (A in m2). We want to check the target temperatures of the six process streams and calculate the hot and cold utility and total annual cost (TAC) of the network. We first calculate the 13  13 coefficient matrix V using Eqs. (6.3), (6.16), which yield its nonzero elements vij as follows: E1: E2: E3: E4: E5: E6: M1:

v1,1 ¼ 0.035474, v3,3 ¼ 0.085687, v5,5 ¼ 0.066855, v7,7 ¼ 0.333332, v9,9 ¼ 0.623621, v11,11 ¼ 0.352057, v13,13 ¼ 1

H1

500

H2

480

H3 H4 H5

v1,2 ¼ 0.964526, v3,4 ¼ 0.914313, v5,6 ¼ 0.933145, v7,8 ¼ 0.666668, v9,10 ¼ 0.376379, v11,12 ¼ 0.647943,

v2,1 ¼ 0.823736, v2,2 ¼ 0.176264, v4,3 ¼ 0.839243, v4,4 ¼ 0.160757, v6,5 ¼ 0.846172, v6,6 ¼ 0.153828, v8,7 ¼ 0.887667, v8,8 ¼ 0.112333, v10,9 ¼ 0.837552, v10,10 ¼ 0.162448, v12,11 ¼ 0.432558, v12,12 ¼ 0.567442,

E1:65.899 m2

E6:8.61 m2

320

(6)

E2:30.653 m2

380

(4)

E3:53.644 m

460

320

2

360

(6)

E5:21.957 m2

380

300 360

(20)

380

E4:39.373 m2

320

M1

290

(12)

700 660

(18)

700

(4.3578)

(9.0124)

(6.6167)

Fig. 6.1 Rating example for a heat exchanger network of Toffolo (2009).

C1

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Design and operation of heat exchangers and their networks

The supply temperature vector of the network is given by T0 ¼ ½ 500 480 460 380 380 290 T The nonzero elements of the four matching matrices are calculated according to the given thermal capacity rates in the channels: g2, 13 ¼ g4,13 ¼ g6, 13 ¼ g10,12 ¼ g11,1 ¼ 1, g13,8 ¼ 0:500689, g13, 10 ¼ 0:499311 0 0 0 0 0 0 0 ¼ g3,2 ¼ g5,3 ¼ g7, g1,1 5 ¼ g8, 6 ¼ g9,4 ¼ g12, 6 ¼ 1 00 00 00 00 00 00 00 g1,11 ¼ g2, 3 ¼ g3, 5 ¼ g4,9 ¼ g5,7 ¼ 1, g6, 2 ¼ 0:390306, g6,4 ¼ 0:2421, 00 ¼ 0:367594 g6,6

Because there is no bypass from entrances to exits of the network, the 000 bypass matrix G ¼ 0. The inlet and outlet temperature vectors of the exchangers and the mixer as well as the exit temperature vector of the network can then be calculated with Eqs. (6.21), (6.13), (6.22), T0E ¼ ½ 500 370:63 480 370:63 460 370:63 380 290 380 326:86 375:22 290 370:63T T00E ¼ ½ 375:22 477:2 380 462:42 376:6 446:25 320 369:89 360 371:37 320 326:86 370:63T T00 ¼ ½320 380 376:60 360 320 462:24T Since the exit temperatures of the hot stream H3 and cold stream C1 have not yet reached their target values, the hot utility and cold utility shall be applied to them, respectively. The heating and cooling loads and heat transfer areas of the heater HUC1 and cooler H3CU are calculated as follows: Heater HUC1:  00  QHUC1 ¼ C_ C1 tC1  t600 ¼ 18  ð660  462:24Þ ¼ 3559 kW 0   00  00 tHU  tC1  tHU  t600 Δtm, HUC1 ¼ 0 00 Þ=ðt 00  t 00 Þ ln ½ðtHU  tC1 HU 6 ð700  660Þ  ð700  462:24Þ ¼ 110:95 K ¼ ln ½ð700  660Þ=ð700  462:24Þ AHUC1 ¼ QHUC1 =ðkΔtm,HUC1 Þ ¼ 3559=ð1  110:95Þ ¼ 32:08 m2 CU,HUC1 ¼ 140QHUC1 ¼ 4:983  105 $=yr, CE, HUC1 ¼ 1200A0:6 HUC1 ¼ 9615$=yr

Optimal design of heat exchanger networks

239

Cooler H3CU:   00 QH3CU ¼ C_ H3 t300  tH3 ¼ 6  ð376:60  360Þ ¼ 99:6 kW  00 00   00  0 t3  tCU  tH3  tCU Δtm, H3CU ¼ 00 Þ  ðt 00  t 0 Þ ln ½ðt300  tCU H3 CU ð376:60  320Þ  ð360  300Þ ¼ 58:28 K ¼ ln ½ð376:60  320Þ=ð360  300Þ AH3CU ¼ QH3CU =ðkΔtm,H3CU Þ ¼ 99:6=ð1  58:28Þ ¼ 1:709 m2 CU,H3CU ¼ 10QH3CU ¼ 996$=yr, CE, H3CU ¼ 1200A0:6 H3CU ¼ 1655$=yr Total annual cost: TAC ¼

6 X

1200A0:6 i + CE,H3CU + CU, H3CU + CE, HUC1 + CU,HUC1

i¼1

¼ 570, 764$=yr

6.2 Mathematical model and calculation methods for sizing heat exchanger networks Design of heat exchanger networks refers to two aspects: parameter design (sizing of a heat exchanger network) and structure design (synthesis of a heat exchanger network). Unlike the rating problem, no general explicit solutions are available for sizing heat exchanger networks. For a given network configuration, there might be infinitely many solutions if there is no restriction on the use of the hot and cold utilities. As a result, sizing a heat exchanger network becomes a constrained optimization problem.

6.2.1 Matrix formulation In the matrix formulation, we express the task of sizing a heat exchanger network as follows: For given supply temperatures of N 0 process streams entering the network 0 T T0 ¼ ½ t10 t20 ⋯ tN 0

240

Design and operation of heat exchangers and their networks

upper and lower bounds of the target temperatures of N 00 process streams leaving the network  00 T 00  00 T 00 00 T00ub ¼ tub , Tlb ¼ tlb, 1 tlb00 , 2 ⋯ tlb00 , N 00 , 1 tub, 2 ⋯ tub, N 00 thermal capacity rates of process streams at N 0 network entrances and N 00 network exits 0 C_ i0 ði0 ¼ 1, 2, …, N 0 Þ,

00 C_ i00 ði00 ¼ 1, 2, …, N 00 Þ

thermal capacity rates of hot and cold streams and overall heat transfer coefficients of NE heat exchangers C_ E, h, j , C_ E, c, j , kj ðj ¼ 1, 2, …, NE Þ utility temperatures of NHU hot utilities and NCU cold utilities 0 00 tHU , k , tHU, k ðk ¼ 1, 2, …, NHU Þ,

0 00 tCU , l , tCU, l ðl ¼ 1, 2, …, NCU Þ

investment costs of process heat exchangers, heaters, and coolers CE ðAÞ, CE, HU, k ðAÞ ðk ¼ 1, 2, …NHU Þ and CE, CU, l ðAÞ ðl ¼ 1, 2, …NCU Þ hot and cold utility costs CU, HU, k ðQÞ ðk ¼ 1, 2, …NHU Þ,

CU, CU, l ðQÞ ðl ¼ 1, 2, …NCU Þ

And the network configuration (excluding the heaters and coolers), together with a set of additional equality and inequality constraints g(x) and h(x), determine the heat transfer areas and thermal capacity rates of NE process heat exchangers, so that the objective function f (x) (usually the total annual cost TAC) reaches the minimum. This can be formulated as follows: min f ðxÞ s:b: T

00

ðxÞ  T00ub

(6.23) 0

(6.24)

T00lb  T00 ðxÞ  0

(6.25)

hðxÞ  0

(6.26)

gðxÞ ¼ 0

(6.27)

The vector x is a set of variables to be optimized, mainly the heat transfer areas and splitting factors of the thermal capacity rates.

Optimal design of heat exchanger networks

241

We first guess the heat transfer areas of all process heat exchangers in the given network. If the heaters and coolers are located at the exits of the network, they will not be included in the matrix formulation. Otherwise, they shall be treated as the process heat exchangers. According to the given network, if there are stream splitting and rejoining, the corresponding splitting factors shall be guessed, following the mass balance constraint: n X

ck ¼ 1

(6.28)

k¼1

where n is the number of splits of a splitting and ck is the splitting factor of the kth split. Luo et al. (2009) suggested that the constraints (6.24) and (6.25) can be treated by adding additional heaters and coolers and taking the corresponding costs as the penalty functions. Using the guessed heat transfer areas and splitting factors, the coefficient matrix V and four matching matrices can be determined, and the outlet temperatures of each exchanger in the network and the exit temperatures of the streams before entering the heaters and coolers can be obtained by the use of the general solution introduced in Section 6.1. After the stream temperatures have been obtained, the constraints (6.24) and (6.25) will be checked. If the exit temperature of a stream is higher than the upper bound of its target value, the stream will be cooled by a cold utility. If it is lower than the lower bound of the target value, the stream will be heated by a hot utility. With this method, the heat exchanger network is always feasible. As an example, we take the total annual cost as the objective function: TAC ¼

NE X   CE AE, j j¼1

+

N 00 X

min fCE, HU, k ðAHU, i00 k Þ + CU, HU, k ðQHU, i00 k Þ; k ¼ 1, …, NHU g

i00 ¼1 N 00 X + ½ min fCE, CU, l ðACU, i00 l Þ + CU, CU, l ðQCU, i00 Þ; l ¼ 1, …, NCU g i00 ¼1

(6.29) in which

n   o 00 QHU, i00 ¼ max C_ i00 tlb00 , i00  ti0000 , 0

(6.30)

242

Design and operation of heat exchangers and their networks

n   o 00 00 QCU, i00 ¼ max C_ i00 ti0000  tub 00 ,i , 0 AHU, i00 k ¼

(6.31)

QHU, i00 FHU, i00 k kHU, i00 k max fΔtm, HU, i00 k ,  Δtm, HU, i00 k =φg

(6.32)

QCU, i00 FCU, i00 l kCU, i00 l max fΔtm, CU, i00 l ,  Δtm, CU, i00 l =φg

(6.33)

ACU, i00 l ¼

 8  0 00 00 00   >  t tHU  t  t 00 00 > > 00 00 00 > 0 < h , k lb, i HU, k i i, t 0 t  t  t 00 00 HU, k HU, k lb, i i 00 00 00 Δtm, HU, i00 k ¼ ln t 0 HU, k  tlb, i00 = tHU, k  ti00 >   > > : 0 00 00 00  0 t  t  t 1, tHU 00 00 HU, k ,k lb, i i (6.34)  8  0   >  t 00 00  tCU t0000  t 00 > ,l > 00 00 0 < hi CU, l ub, i i, ti0000  tCU tub >0  tCU ,l ,l , i00 00 0 Δtm, CU, i00 l ¼ ln t0000  t 00 = t  t 00 CU, l CU, l i ub, i >   > > : 00 00 0 1, ti0000  tCU tub 0  tCU ,l ,l , i00 (6.35)

CE(A) and CU(Q) are investment cost function and utility cost function. They are usually expressed as CE ðAÞ ¼ a + bAn

(6.36)

CU ðQÞ ¼ cQ

(6.37)

where a, b, and c are cost constants. φ is the penalty factor against negative Δtm. It is a large positive value, for example, 103, yielding a much larger heat transfer area. Applying a constrained optimization algorithm to the sizing problem (6.23) under the constraints (6.26) and (6.27), we can determine the optimal heat transfer area of each exchanger in the network together with the optimal thermal capacity rates of stream splits. Example 6.2 Optimal sizing of the heat exchanger network. Example 6.1 will be used here for the optimal sizing problem. We set the heat transfer areas of the six process heat exchangers AE1–AE6 and the thermal capacity rates of cold stream C1 in exchangers C_ c, E2 –C_ c,E4 as the variables to be optimized:  T x ¼ AE1 AE2 AE3 AE4 AE5 AE6 C_ c,E2 C_ c,E3 C_ c,E4

Optimal design of heat exchanger networks

243

According to the mass balance constraints, we have C_ c,E1 ¼ C_ C1  C_ c, E2  C_ c, E3 , C_ c,E5 ¼ C_ C1  C_ c, E4 TAC is taken as the objective function TACðxÞ ¼

6 X j¼1

1200A0:6 E, j +

6  X

0:6 1200A0:6 HU,i + 140QHU, i + 1200ACU, i + 10QCU,i



i¼1

in which the areas and heat loads of heaters and coolers are calculated by Eqs. (6.30)–(6.35) according to the calculated exit stream temperatures of the network (excluding utilities) for given x by means of the matrix method. The variable vector x is optimized by the use of Excel solver. The upper and lower bounds of the variables are set to be [1, 100] for the heat transfer areas and [1, 10] for the thermal capacity rates, and their lower bounds are used as the initial values for the optimization. Besides the upper and lower bounds of the variables, two mass balance constraints C_ c,E1  0 and C_ c, E2  0 are given in the solver. The penalty factor φ ¼ 1000 is used for Eqs. (6.32), (6.33). By repeated use of the evolutionary solving method EA of the solver, the design is optimized and converges to TAC ¼ 570,777$/yr, which is very close to the global optimization result, TAC ¼ 570,764$/yr (see Example 6.1). The calculation results are listed in Table 6.1.

6.2.2 Nonlinear programming formulation In the nonlinear programming (NLP) formulation, we express the task of sizing a heat exchanger network as follows: For given supply temperatures, upper and lower bound target temperatures and thermal capacity rates of N process streams 00 00 _ ti0 , tub , i , tlb, i , C i ði ¼ 1, 2, …, N Þ

thermal capacity rates of hot and cold streams and overall heat transfer coefficients of NE heat exchangers C_ h, j , C_ c, j , kj ðj ¼ 1, 2, …NE Þ inlet and outlet temperatures of NHU hot utilities and NCU cold utilities 0 00 tHU , k , tHU, k ðk ¼ 1, 2, …NHU Þ,

0 00 tCU , l , tCU, l ðl ¼ 1, 2, …NCU Þ

investment costs of process heat exchangers, heaters, and coolers CE ðAÞ, CE, HU, k ðAÞ ðk ¼ 1, 2, …NHU Þ,

CE, CU, l ðAÞ ðl ¼ 1, 2, …NCU Þ

244

th0 (K) th00 (K)

C_ h (kW/K) tc0 (K) tc00 (K) C_ c (kW/K) Δtm (K) A (m2) Q (kW) CE ($/yr) CU ($/yr)

E1

E2

E3

E4

E5

E6

HUC1

H3CU

500 375.631 6 370.766 477.038 7.022 11.662 63.984 746.213 14,549 –

480 380 4 370.766 462.970 4.338 12.737 31.405 400.000 9492 –

460 376.661 6 370.766 446.072 6.640 9.343 53.522 500.036 13,071 –

380 320 12 290 369.872 9.014 18.300 39.344 720.000 10,867 –

380 360 20 327.147 371.663 8.986 17.877 22.375 400.000 7745 –

375.631 320 6 290 327.147 8.986 38.505 8.669 333.787 4385 –

700 700 – 462.224 660 18 110.957 32.084 3559.964 9615 498,395

376.661 320 6 300 320 – 58.314 1.714 99.964 1658 1000

Design and operation of heat exchangers and their networks

Table 6.1 Optimal sizing of the heat exchanger network of Toffolo (2009).

245

Optimal design of heat exchanger networks

hot and cold utility costs CU, HU, k ðQÞ ðk ¼ 1, 2, …NHU Þ,

CU, CU, l ðQÞ ðl ¼ 1, 2, …NCU Þ

and the network configuration (excluding the heaters and coolers) together with a set of additional equality and inequality constraints g(x) and h(x) determine the heat loads and thermal capacity rates of process heat exchangers QE, j , C_ E, h, j , C_ E, c, j ðj ¼ 1, 2, …NE Þ so that the objective function f (x) (usually the total annual cost) reaches the minimum. As an example, we take the total annual cost as the objective function minf ðxÞ ¼ TAC ¼

NE X

  CE AE, j

j¼1

+ +

N X i¼1 N X

min fCE, HU, k ðAHU, ik Þ + CU, HU, k ðQHU, i Þ; k ¼ 1, 2, …, NHU g

min fCE, CU, l ðACU, il Þ + CU, CU, l ðQCU, i Þ; l ¼ 1, 2, …, NCU g

i¼1

(6.38) in which

  QHU, i ¼ max C_ i tlb00 , i  ti00 , 0

(6.39)

  00 QCU, i ¼ max C_ i ti00  tub ,i , 0

(6.40)

AE, k ¼ AHU, ik ¼

max fQE, k ,  φQE, k g FE, k kE, k max fΔtm, E, k ,  Δtm, E, k =φg

(6.41)

max fQHU, i ,  φQHU, i g FHU, ik kHU, ik max fΔtm, HU, ik ,  Δtm, HU, ik =φg

(6.42)

max fQCU, i ,  φQCU, i g FCU, il kCU, il max fΔtm, CU, il ,  Δtm, CU, il =φg

(6.43)

ACU, il ¼

 8    > th0 , E, k  tc00, E, k  th00, E, k  tc0 , E, k > > < h  i , th0 , E, k  tc00, E, k th00, E, k  tc0 , E, k > 0 00 00 0 Δtm, E, k ¼ ln t 0 h, E, k  tc, E, k = th, E, k  tc, E, k >   > > : 1, th0 , E, k  tc00, E, k th00, E, k  tc0 , E, k  0 (6.44)

246

Design and operation of heat exchangers and their networks

 8  0 00 00 00   >  t tHU  t  t > i > 0 00 00 00 < h , k lb, i HU, k i , tHU t >0  t  t i lb, i HU, k ,k 0 00 00 00 Δtm, HU, ik ¼ ln tHU = t  t  t i lb HU , k , i , k >   > > : 0 00 00 00 1, tHU  t  t t 0 i lb, i HU, k ,k (6.45)

 8  00 00  t 0   > ti00  tCU  tub > CU , l , i , l > 00 00  t 0 < h  i , ti00  tCU t ub, i CU, l > 0 ,l 00 00  t 0 Δtm, CU, il ¼ ln ti00  tCU = t ub, i CU, l ,l >   > > : 1, 00 00  t 0 ti00  tCU tub CU, l  0 ,l ,i

(6.46)

in which the indices “i” the ith process stream, “j” the jth process heat exchanger, “k” the kth hot utility and, “l” the lth cold utility. φ is the penalty factor against negative Q and negative Δtm. A large value of φ, for example, 103, will yield a much larger heat transfer area. The constraints applied to Eq. (6.38) consist of mass balance and energy balance constraints for mixers, process heat exchangers, heaters, coolers and the whole network, and the additional equality and inequality constraints g(x) and h(x). To set up the relations between different constraint groups, the necessary mapping tables are introduced for the formulation. For example, the mapping table for the inlet of the NM mixers set the pointer of the n0 th coming stream of the mth mixer to be the pointer of one of the five sources (the ith stream at network entrance, hot stream out let of the jth heat exchanger, cold stream out let of the jth heat exchanger, or the n00 th outlet of the m00 th mixer). Similar mapping tables can be set for process heat exchangers and network exits (excluding the utilities). These mapping tables constitute a set of equality constraints. Other constraints are as follows: (1) Mixers We define a node as a mixer if NM,in process streams are mixed at the node, or the stream leaving the node is splitted into NM,out substreams. NM is the number of the mixers in the network. Mass balance constraints 0 NM ,m

X n0 ¼1

0 C_ M, mn0

00 NM ,m



X

n00 ¼1

00 C_ M, mn00 ¼ 0 ðm ¼ 1, 2, …NM Þ

(6.47)

Energy balance constraints 00 tM , mn00

0 NM ,m

X n0 ¼1

0 C_ M, mn0

0 NM ,m



X n0 ¼1

0 0 C_ M, mn0 tM , mn0

  00 ¼ 0 m ¼ 1, 2, …, NM ; n00 ¼ 1, 2, …, NM ,m

(6.48)

247

Optimal design of heat exchanger networks

(2) Process heat exchangers Energy balance constraints   C_ E, h, j tE0 , h, j  tE00, h, j  QE, j ¼ 0, C_ E, c, j tE00, c, j  tE0 , c, j  QE, j ¼ 0 ðj ¼ 1, 2, …NE Þ

(6.49)

Thermodynamics constraints tE00, c, j  tE0 , h, j  0, tE0 , c, j  tE00, h, j  0 ðj ¼ 1, 2, …NE Þ

(6.50)

(3) Additional constraints hðxÞ  0

(6.51)

gðxÞ ¼ 0

(6.52)

Applying a constrained optimization algorithm to the sizing problem (6.38) under the mapping constraints and the constraints (6.47)–(6.52), we can determine the optimal heat load of each exchanger in the network together with the optimal thermal capacity rates of stream splits. Example 6.3 Sizing a heat exchanger network. This example is the best network structure for the synthesis problem taken from Yee and Grossmann (1990). As is shown in Fig. 6.2, the network has two hot process streams and two cold process streams. There are four process heat exchangers E1–E4, two mixers M1 and M2. S1 and S2 denote the splits of stream C1. The set of stream is defined as [H1, H2, C1, C2]. We use the NLP formulation to solve the sizing problem and take the TAC as the objective function TACðxÞ ¼

4 X j¼1

H1 H2

443

1000A0:6 E, j +

4  X

0:6 1200A0:6 HU,i + 80QHU, i + 1000ACU, i + 20QCU,i

i¼1

E1

E2

E3

293

(30)

E4

423

353

(15)

408

M2

S1

M1

333 (20)

413

S2

303 (40)

Fig. 6.2 Sizing example for the heat exchanger network for H2C2_443K.

C1 C2



248

Design and operation of heat exchangers and their networks

in which the variables to be optimized is set to be  T x ¼ QE1 QE2 QE3 QE4 C_ c, E4 According the network structure shown in Fig. 6.2, the equality constraints can be expressed as Heat load :

QE, j ¼ xj ðj ¼ 1, 2, 3, 4Þ

Hot stream :

C_ E,h, 1 ¼ C_ E,h, 2 ¼ C_ E, h,3 ¼ C_ 1 , C_ E,h, 4 ¼ C_ 2

Inlet :

0 0 0 00 0 00 0 0 tE,h, 1 ¼ t1 , tE, h,2 ¼ tE,h, 1 , tE,h,3 ¼ tE, h,2 , tE,h, 4 ¼ t2

Outlet :

00 0 _ tE,h, j ¼ tE, h, j  QE, j =C E,h, j ðj ¼ 1, 2, 3, 4Þ

Cold stream :

00 00 C_ E,c,1 ¼ C_ M,2,1 , C_ E,c,2 ¼ C_ 4 , C_ E, c, 3 ¼ C_ M, 1,1 , 00 C_ E,c,4 ¼ C_ M,1,2

Inlet :

0 00 0 0 0 00 0 00 tE, c, 1 ¼ tM, 2,1 , tE,c,2 ¼ t4 , tE,c,3 ¼ tM, 1,1 , tE,c,4 ¼ tM, 1,2

Outlet :

00 0 _ tE,c, j ¼ tE, c, j + QE, j =C E, c, j ðj ¼ 1, 2, 3, 4Þ

Mixer inlet :

0 0 0 C_ M,1,1 ¼ C_ 3 , C_ M, 2,1 ¼ C_ E, c, 3 , C_ M,2,2 ¼ C_ E,c,4 0 0 0 00 0 00 tM, 1,1 ¼ t3 , tM,2, 1 ¼ tE, c, 3 , tM,2, 2 ¼ tE, c, 4

Mixer outlet :

00 0 00 00 C_ M,1,1 ¼ C_ M,1, 1  C_ M,1, 2 , C_ M, 1,2 ¼ x5 , 00 0 0 00 00 0 ¼ tM, C_ M,2,1 ¼ C_ M,2, 1 + C_ M, 2,2 , tM,1,1 1, 2 ¼ tM,1,1 ,   00 0 0 _0 _0 _0 _0 tM,2, 1 ¼ C M,2,1 tM,2,1 + C M, 2, 2 tM, 2, 2 = C M,2,1 + C M, 2, 2

00 00 00 00 00 00 00 Network exit : t100 ¼ tE, h,3 , t2 ¼ tE,h, 4 , t3 ¼ tE,c,1 , t4 ¼ tE,c,2

Solving the earlier equality constraint system, we can obtain the outlet temperatures of the process heat exchangers, mixers, and the exit temperatures on the network. However, the guessed heat loads might cause an unrealistic temperature cross 0   th  tc00 th00  tc0  0 Such cases can be avoided by the use of the unequality constraints (6.50) or by the use of the penalty factor in Eqs. (6.41)–(6.46). The optimization was carried out with the Excel solver. The upper and lower bounds of the variables were set to be [1, 5000] for the heat transfer areas and [1, 19] for the thermal capacity rates, and their lower bounds were used as the initial values for the optimization. Besides the upper and lower bounds of the variables, Eq. (6.50) was also set as the unequality constraints in the solver. The penalty factor φ ¼ 1000 was used for Eqs. (6.41)–(6.43). By repeated use of the evolutionary solving method EA of the solver, we obtained the best design for the H2C2_443K problem, of which the

249

Optimal design of heat exchanger networks

Table 6.2 Optimal sizing of the heat exchanger network for H2C2_443K problem.

th0 (K) th00 (K) C_ h (kW/K) tc0 (K) tc00 (K) C_ c (kW/K) Δtm (K) A (m2) Q (kW) CE ($/yr) CU ($/yr)

E1

E2

E3

E4

H1CU

443 436.151 30 397.727 408 20 36.686 7.001 205.459 3214.281 –

436.151 356.151 30 353 413 40 10.029 299.132 2400 30,585.663 –

356.151 346.333 30 293 353.675 4.854 16.566 22.224 294.541 6428.335 –

423 303 15 293 411.846 15.146 10.566 212.939 1800 24,943.272 –

346.333 333 30 293 313 – 36.565 13.674 400 4803.299 8000

TAC ¼ 77,975$/yr, which is very close to the global optimized TAC of 77,964$/yr (see Example H2C2_443K in Section 6.4). The calculation results are listed in Table 6.2.

6.3 Pinch technology for synthesis of heat exchanger networks A more complicated task is the structure design of heat exchanger networks (synthesis problem). A fundamental synthesis problem of a heat exchanger network can be stated as follows. For given Nh hot streams to be cooled, Nc cold streams to be heated, NHU hot utilities and NCU cold utilities available for heating and cooling the process streams, configure a heat exchanger network that has the minimum total annual cost (TAC, the sum of annual costs of process heat exchangers, heaters and coolers, and annual costs of hot and cold utilities) under a set of constraints such as target stream temperatures: X   X minTACðxÞ ¼ CE AE, j + ½CE, H, k ðAE, H, k Þ + CU, H, k ðQH, k Þ j

+

X

k

½CE, C, l ðAE, C, l Þ + CE, C, l ðQC, l Þ

k

(6.53)

250

Design and operation of heat exchangers and their networks 00 00 00 s:b: th00, i  tub , h, i  0, tlb, h, i  th, i  0 ði ¼ 1, 2, …, NH Þ 00 00 00 tc00, j  tub , c, j  0, tlb, c, j  tc, j  0 ð j ¼ 1, 2, …, NC Þ

(6.54) (6.55)

and additional inequality and equality constraints: hðxÞ  0

(6.56)

gðxÞ ¼ 0

(6.57)

The annual investment costs can be calculated in two ways. The one is for using available invested capital, for which the annuity factor is calculated based on the capital appreciation in financial markets during the plan lifetime n: Ca ¼ ð1 + r Þn =n

(6.58)

The other is for using loan, for which it is calculated based on the fixed annual repayment of the loan during the plan lifetime: Ca0 ¼

r0 1  ð1 + r 0 Þn

(6.59)

In the synthesis problem, the possible structures of a heat exchanger network could be astronomical figures, and the traditional optimization solvers cannot be directly applied to Eqs. (6.53)–(6.57). Therefore, lots of synthesis methodologies have been developed, and the most practical procedure is the pinch technology. The pinch technology is the first complete practical method for the synthesis of heat exchanger networks. In the late 1970s, the pinch design method was developed by Linnhoff and his coworkers (Linnhoff and Flower, 1978a,b; Linnhoff et al., 1979, 1982; Flower and Linnhoff, 1980; Linnhoff and Hindmarsh, 1983). In the pinch technology, the minimization of the energy usage is addressed from a thermodynamic point of view. At first, the minimum requirement of heating and cooling of the process streams through a heat cascading calculation is determined. This calculation is executed before the design of the equipment. Then, the method identifies a location of the minimum temperature difference in the process called “pinch point.” The pinch represents the most constrained location of a design. It divides the overall problem into two independent problems that are further treated separately. This means that the design starts at the pinch and is carried out from it toward two opposite directions, like two separates problems. At the pinch, quite often, there is a crucial or essential match.

Optimal design of heat exchanger networks

251

If this match is not satisfied, the utility usage increases, and additional penalties may result during the remaining design. The identification of the essential and other matches at the pinch is achieved by applying feasibility criteria to the stream data at the pinch. Also, the feasibility criteria can indicate whether it is necessary for a stream splitting. After leaving away from the pinch, the design task is no longer so constraint. The main synthesis steps of the pinch design method are as follows: (1) assume the minimum temperature difference Δtm and build the problem table, (2) draw the composite curves, (3) match the hot and cold process streams following the pinch design principles and rules, and (4) optimize Δtm and repeat the earlier steps until the pinch position does not change. The key established principles of the pinch technology are demonstrated in detail in Linnhoff et al. (1982).

6.3.1 Problem table The problem table algorithm proposed by Linnhoff and Flower (1978a) is used to determine the pinch location and was formulated by Luo and Roetzel (2010, 2013). For a given synthesis task dealing with Nh hot streams and Nc cold streams and for a specified value of the minimum temperature difference Δtmin, we define the following temperature vectors: 2 0 3 2 00 3 2 0 3 2 0 3 tc, 1 + Δtmin tc∗ , 1 th, 1 th, 1 6 t0 7 6 00 7 6 t0 7 6 t0 + Δtmin 7 6 h, 2 7 00 6 th, 2 7 0 6 c∗ , 2 7 6 c, 2 7 T0h ¼ 6 7, Th ¼ 6 7, Tc∗ ¼ 6 7¼6 7, 4 ⋮ 5 4 ⋮ 5 4 ⋮ 5 4 5 ⋮ th0 , Nh 2 00 tc∗ , 1 6 t00 6 c∗ , 2 T00c∗ ¼ 6 4 ⋮ tc00∗ , Nc

3

2

th00, Nh tc00, 1 + Δtmin

7 6 t00 + Δtmin 7 6 c, 2 7¼6 5 4 ⋮

3

tc0 ∗ , Nc

tc0 , Nc + Δtmin

7 7 7 5

tc00, Nc + Δtmin (6.60)

Let the set n o n o n o ST ¼ th0 , 1 , th0 , 2 , …, th0 , Nh [ th00, 1 , th00, 2 , …, th00, Nh [ tc0 ∗ , 1 , tc0 ∗ , 2 , …, tc0 ∗ , Nc n o [ tc00∗ , 1 , tc00∗ , 2 , …, tc00∗ , Nc (6.61)

252

Design and operation of heat exchangers and their networks

then we can further define a temperature level vector T ¼ ½t1 t2 …tNSN + 1 T

(6.62)

in which the temperature levels ti 2 ST (i ¼ 1, 2, …, NSN + 1) and t1 > t2 > … > tNSN+1. The streams in each temperature interval [ti, ti+1] consists a subnetwork SNi (i ¼ 1, 2, …, NSN). The heat transport difference between the heat input Ii and heat output Oi in SNi can be calculated by means of Eq. (6.63): Di ¼ Ii  Oi ¼ ΔHc, i  ΔHh, i

(6.63)

in which ΔHh,i and ΔHc,i are the total enthalpy change of hot streams and that of the cold streams in the subnetwork SNi ΔHh, i ¼ ðti  ti + 1 Þ

Nh X

C_ h, ij

(6.64)

C_ c, ij

(6.65)

j¼1

ΔHc, i ¼ ðti  ti + 1 Þ C_ h, ij ¼ C_ c, ij ¼





Nc X j¼1

C_ h, j , th00, j  ti + 1 and ti  th0 , j 0, others

(6.66)

C_ c, j , tc0 ∗ , j  ti + 1 and ti  tc00∗ , j 0, others

(6.67)

According to the energy balance, if there is no heat utility connecting to SNi+1, the heat input of SNi+1 should be equal to the heat output of SNi: I i + 1 ¼ Oi

(6.68)

We begin the calculation of heat input Ii from SN1, assuming I1 ¼ 0, to that of the last subnetwork INSN. The assumption of I1 ¼ 0 might yield negative values of heat inputs and heat outputs of the subnetworks. This is not allowed because the heat cannot flow from a lower temperature region to a higher temperature region. Therefore, a modification should be performed by adding the minimum hot utility duty QHU, min ¼ min fIi , Oi ; i ¼ 1, 2, …, NSN g

(6.69)

to all heat inputs and outputs. After the modification, we also obtain the minimum cold utility duty QCU, min ¼ ONSN

(6.70)

Optimal design of heat exchanger networks

253

The position where the heat input is zero is the pinch. The problem table provides a simple framework for numerical analysis. For simple problems, it can be quickly evaluated by hand. For larger problems, it is easily implemented on the computer. With the problem table algorithm, the engineer has a powerful targeting technique at their fingertips. Data can be quickly extracted from flow sheets and analyzed to see whether the process is nearing optimal or whether significant scope for energy saving exists.

6.3.2 Composite curves The composite curves were first used by Huang and Elshout (1976). A similar concept is the composite line in the available energy diagram applied by Umeda et al. (1978). To structure the composite curves, we draw the hot and cold process streams on a temperature-heat content (enthalpy) diagram. Starting from the individual streams, one composite curve is constructed for all hot streams in the process and another for all cold streams by simple addition of heat contents over the temperature ranges in the problem. The overlap between the two composite curves represents the maximum amount of heat recovery possible within the process. The “overshoot” of the hot composite curve represents the minimum amount of required external cooling duty, and the “overshoot” of the cold composite curve represents the minimum amount of required external heating duty. To draw the composite curves, let j1 and j2 indicate the maximum and minimum temperature levels of the hot streams; k1 and k2 indicate those of the cold streams, respectively, so that n o n o tj1 ¼ max th0 , 1 , th0 , 2 , …, th0 , Nh , tj2 ¼ min th00, 1 , th00, 2 , …, th00, Nh (6.71) n o n o tk1 ¼ max tc00∗ , 1 , tc00∗ , 2 , …, tc00∗ , Nc , tk2 ¼ min tc0 ∗ , 1 , tc0 ∗ , 2 , …, tc0 ∗ , Nc (6.72) Then, we can calculate the enthalpy flow rates of the hot and cold streams at the temperature levels tj with Eq. (6.73): Hh, j ¼

j2 1 X i¼j

ΔHh, i , Hc, k ¼

kX 2 1

ΔHc, i + QCU, min

(6.73)

i¼k

to get the coordinates of the composite curve of the hot streams (tj, Hh, j) (j ¼ j1, j1 + 1, …, j2) and those of the cold streams (tk – Δtm, Hc,k) (k ¼ k1, k1 + 1, …, k2), respectively, and draw the polyline composite curves.

254

Design and operation of heat exchangers and their networks

180 QHU,min

160 140 120 t (°C)

Dtm 100 80 60 40 20 0

QCU,min 0

1000

2000 H (kW)

3000

4000

Fig. 6.3 Composite curves of Example 6.4 (H2C2_175R), Δtm ¼ 5 K.

Because of the polyline nature of the composite curves, they approach to each other most closely at the pinch point where Δtmin occurs. The composite curves allow the designer to predict optimized hot and cold utility targets ahead of designing the network. Taking the composite curves shown in Fig. 6.3 as an example, if we move the cold composite curve to the right, Δtmin will increase, which means a decrease in the heat exchanger area, that is, a decrease in the investment costs. However, both hot and cold utility costs will increase. If we move the cold composite curve to the left until Δtmin approaches to zero, it is rather the opposite. An experienced engineer can make a good balance between the investment costs and utility costs to get a good design of the network.

6.3.3 Pinch design method According to the composite curves, we can separate the network at the pinch into two sections. In the section above the pinch (hot end part), the composite hot stream gives all its heat to the composite cold stream with only residual heating required. The system is therefore a heat sink. Heat flows from the hot utility into the network, but no heat leaves it. Conversely, below the pinch (cold end part), the system is a heat source. Heat

Optimal design of heat exchanger networks

255

in transferred from the system to the cold utility, but no heat flows into the system. If there is a heat flow across the pinch, the excess external cooling and excess external heating for the same amount of heat are required, which increases both the hot and cold utility costs. The pinch principle allows the designer to keep the level of heat recovery optimized by simply making sure that the heat crossing the pinch is minimized. Because any network design that transfers heat across the pinch will cause both heating and cooling duties larger than their minimum, there are three principles: (1) Do not use cold utilities above the pinch. (2) Do not use hot utilities below the pinch. (3) Do not transfer heat across the pinch. For the network design, there are therefore three consequences: (1) Divide the network at the pinch into two parts. (2) Design each part separately. (3) Do not use heaters in the part above the pinch (hot end part); do not use coolers in the part below the pinch (cold end part). For the matching of streams, two rules should be followed: (1) In the part above the pinch, the number of the hot streams (including their branches) should be less than or equal to that of the cold streams (including their branches), that is, Nh  Nc ðabove the pinchÞ

(6.74)

Otherwise, the stream splitting is necessary to ensure that Eq. (6.74) is fulfilled. Similarly, in the part below the pinch, the inequality is inversed: Nh  Nc ðbelow the pinchÞ (6.75) (2) For a match in the part above the pinch, the thermal capacity rate of a hot stream (or the branch of a hot stream) should be less than or equal to that of the cold stream (or the branch of a cold stream) to be matched, that is, C_ h  C_ c ðabove the pinchÞ (6.76) Otherwise, the stream splitting is necessary. For a match in the part below the pinch, the inequality is inversed: C_ h  C_ c ðbelow the pinchÞ The pinch design method is illustrated in detail in Example 6.4.

(6.77)

256

Design and operation of heat exchangers and their networks

In the pinch design method, Δtmin is an important parameter for the balance between the investment costs and utility costs. A large value of Δtmin would decrease the investment costs but increase the utility costs and vice versa. Furthermore, the pinch position could also change with Δtmin. The value of Δtmin can be optimized by taking the total costs of the network as the objective function. The pinch design method focuses on the matches of streams near the pinch because at that point, the temperature difference is the minimum. For the matches away from the pinch, the earlier rules must not be fulfilled. In some cases, there might be multiple pinches or no pinch. A detailed description of the pinch design method can be found in Linnhoff et al. (1982). Example 6.4 Pinch method for H2C2_175R. We take the problem data of H2C2_175R (Ravagnani et al., 2005. See Table 6.3) as an example to illustrate how to design the network with the pinch technology (Luo and Roetzel, 2010, 2013). The problem deals with two hot streams (Nh ¼ 2) and two cold streams (Nc ¼ 2). Let Δtmin ¼ 5 K, it is easy to calculate the problem table by Eqs. (6.63)– (6.70), which gives the pinch position at th ¼ 125°C, QHU,min ¼ 200 kW, QCU,min ¼ 120 kW. Other results are given in Table 6.4. The composite curves are shown in Fig. 6.3. The detailed calculation procedure can be found in the MatLab code for Example 6.4 in Appendix. To design the network, we divide the problem into two parts at the pinch, as is shown in Fig. 6.4. In the part above the pinch, there is only one match: H1C1, that is, Nh ¼ Nc ¼ 1 with C_ H1 ¼ 10 kW/K, C_ C1 ¼ 20 kW/K; therefore, Eqs. (6.74), (6.76) are fulfilled, and no splitting is necessary. In the part below the pinch, Nh ¼ Nc ¼ 2, which meets the rule (6.75). We would like to choose the matches H1C1 and H2C2 due to their Table 6.3 Problem data for H2C2_175R (Ravagnani et al., 2005). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 HU CU

175 125 20 40 180 15

45 65 155 112 179 25

10 40 20 15

2.615 1.333 0.917 0.166 5 2.5

110 10

Heat exchanger cost ¼ 1200A0.57$/yr (A in m2)

Table 6.4 Problem table for H2C2_175R (Ravagnani et al., 2005), Δtmin ¼ 5 K. t, th (°C)

ΔHc (kW)

ΔHh (kW)

D (kW)

I (kW)

O (kW)

Hh (kW)

tc (°C)

Hc (kW)

1 2 3 4 5 6 7

175 160 125 117 65 45 –

0 700 160 1820 700 400 –

150 350 400 2600 200 0 –

150 350 240 780 500 400 –

200 350 0 240 1020 520 –

350 0 240 1020 520 120 –

3700 3550 3200 2800 200 0 –

– 155 120 112 60 40 20

– 3900 3200 3040 1220 520 120

Optimal design of heat exchanger networks

SN

257

258

Design and operation of heat exchangers and their networks

H1

45°C

125°C

C1 C2

175°C (10 kW/K)

65°C H2

125°C

(10 kW/K)

125°C

(40 kW/K)

20°C

120°C

H2 120°C

155°C

(20 kW/K)

(20 kW/K)

40°C

H1

C1

112°C

(15 kW/K)

Cold end part (below the pinch)

Pinch

Hot end part (above the pinch)

Fig. 6.4 Pinch decomposition of Example 6.4 (H2C2_175R), Δtm ¼ 5 K.

temperature intervals. Since C_ H1 < C_ C1 , according to Eq. (6.77), a splitting in C1 with a new match H2C1 and a corresponding splitting in H2 is necessary. Furthermore, as has been analyzed in the aforementioned problem table, the minimum cooling duty for hot stream H1 is 120 kW, and the minimum heating duty for cold stream C1 is 200 kW; therefore, a cooler is added to hot stream H1, and a heater is added to cold stream C1. Thus, we obtain a network configuration shown in Fig. 6.5. To determine the thermal capacity rates of the splits, we assume at first that the mixing of the splits is isothermal. According to the mass and energy balance constraints, we have  00  0  0 00 C_ h, E4 ¼ C_ c, 2 tc,2 = th,2  th,2 ¼ 18 kW=K, C_ h,E3 ¼ C_ h, 2  C_ h,E4  tc,2 ¼ 22 kW=K 0    00 0  th,2 = tc,pinch  tc,1 ¼ 13:2 kW=K, C_ c, E3 ¼ C_ h,E, 3 th,2 _ _ _ C c, E2 ¼ C c, 1  C c,E3 ¼ 6:8 kW=K

H1

H2

175

E1

tpinch

E2

45

(10)

E3

123

65

(40)

155

20 (20)

112

40 tpinch – Dtm

(15)

E4

Fig. 6.5 Network configuration of Example 6.4 (H2C2_175R).

C1 C2

Optimal design of heat exchanger networks

Now, the two rules for the matching of streams will be checked. In the part below the pinch, Nh ¼ Nc ¼ 3, and since C_ h, E2 ¼ 10 kW/K and C_ c,E4 ¼ 15 kW/K, all the three matchings fulfills with Eq. (6.77). For Δtmin ¼ 5 K, the pinch is located at th ¼ 125°C and TAC ¼ 109,768$/yr. As has been mentioned earlier, for given investment and utility cost functions, there is an optimal value of Δtmin ¼ 4.055 K, yielding the minimum TAC of 109,535$/yr. The calculation results are provided in Table 6.5. Analyzing the heat transfer among the streams without the restricts of isothermal mixing and the third principle of the pinch technology, “Do not transfer heat across the pinch,” we find that the network has four independent variables, two for stream splits (e.g., C_ h, E4 and C_ c,E3 ) and two for heat loads of the process heat exchangers (e.g., QE1 and QE2). Optimizing these four variables, we finally obtain the best network for this design problem with TAC ¼ 108,072$/yr, in which C_ h,E4 ¼ 20:723533 kW=K, C_ c,E3 ¼ 13:032104 kW=K QE1 ¼ 483:13886kW, QE2 ¼ 712:508733kW

Example 6.5 Pinch method for H2C2_150. This example is taken from Zhu (1997). The problem data are listed in Table 6.6. It deals with two hot streams (Nh ¼ 2) and two cold streams (Nc ¼ 2). For Δtmin ¼ 10 K, we obtain that the pinch position is located at th ¼ 90°C, with QHU,min ¼ 5500 kW and QCU,min ¼ 2500 kW. The problem table is given in Table 6.7. The composite curves are shown in Fig. 6.6. It is interesting to notice that the zero heat input I happens at th ¼ 60°C and 90°C, and the hot and cold composite curves are parallel in the cold stream temperature range of 60–90°C, because in this range, the sum of the thermal capacity rates of hot streams H1 and H2 is the same as that of the cold stream C1. To design the network, we divide the problem into two parts at the pinch. In the part above the pinch, there are two hot streams and two cold streams, that is, Nh ¼ Nc with C_ H2 < C_ H1 < C_ C1 < C_ C2 ; therefore, no splitting is necessary. According to the temperature levels of hot and cold streams, Zhu (1997) adopted the matches as H1C2 and H2C1. In the part below the pinch, there are two hot streams but only one cold stream, Nh > Nc, but C_ H2 < C_ H1 < C_ C1 , so a splitting in C1 is required.

259

260

Δtmin

tpinch (°C)

C_ h, E4 (kW/K)

C_ c, E3 (kW/K)

QE1 (kW)

QE2 (kW)

QE3 (kW)

QE4 (kW)

QHU (kW)

QCU (kW)

TAC ($/yr)

5 4 3 4.0550 Optimal

125 125 125 125 design

18 18 18 18 20.724

13.2 13.069 12.941 13.076 13.032

500 500 500 500 483.14

680 700 720 698.90 712.51

1320 1320 1320 1320 1320

1080 1080 1080 1080 1080

200 180 160 181.10 184.35

120 100 80 101.10 104.35

109,768 109,536 109,938 109,535 108,072

Design and operation of heat exchangers and their networks

Table 6.5 Results of pinch design method under different Δtmin (isothermal mixing).

Optimal design of heat exchanger networks

261

Table 6.6 Problem data for H2C2_150 (Zhu, 1997). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 HU CU

150 170 50 80 180 20

50 40 120 110 180 40

200 100 300 500

0.2 0.2 0.2 0.2 0.2 0.2

110 10

Heat exchanger cost ¼ 0.295260  (30,800 + 750A0.81) $/yr (A in m2) (Plant lifetime: 5 years; loan rate of interest: 15%; annualization factor: 0.298) Keeping in mind that C_ h  C_ c should be fulfilled in the part below the pinch, we have to choose C_ c,E3 ¼ 200 kW/K and _ _ _ C c,E4 ¼ C c, 1  C c, E3 ¼ 100 kW/K. Thus, the network is configured as shown in Fig. 6.7. Using the MatLab code in Appendix for Example 6.5, we can determine all the design parameters and the corresponding TAC of 1,819,308$/yr. By optimizing Δtmin, we find that Δtmin ¼ 10.9834 K results in the minimum TAC of 1,816,366$/yr. Of course, this result is obtained under the constraints of isothermal mixing and no heat transfer across the pinch. The network has five independent variables. Further optimization of these variables offers us a better design with C_ c, E3 ¼ 203:39335 kW/K, QE1 ¼ 11,680.172 kW, QE2 ¼ 7814.986 kW, QE3 ¼ 6252.591 kW, QE4 ¼ 2960.676 kW, and TAC ¼ 1,815,294$/yr. However, better network configurations have been found by several researchers in the last 3 years by means of hybrid particle swarm optimization algorithms (Pava˜o et al., 2016; Zhang et al., 2016a,b; Wang et al., 2017). We obtained the best network configuration using hybrid genetic algorithm (Luo et al., 2009), of which TAC ¼ 1,805,242$/yr (see Table 6.10 and Fig. 6.10). The result of pinch design method is only 0.56% higher than that of the best result so far. This example tells us that the pinch design method might not bring us the global optical design solution, but it is really effective and easy to be executed.

6.4 Mathematical programming for synthesis of heat exchanger networks With the development of computer technology, mathematical programming methods were introduced into the synthesis of heat exchanger networks. The network design is defined as optimizing an objective, for

262

SN

t, th (°C)

ΔHc (kW)

ΔHh (kW)

D (kW)

I (kW)

O (kW)

Hh (kW)

tc (°C)

Hc (kW)

1 2 3 4 5 6 7 8

170 150 130 120 90 60 50 40

0 0 3000 24,000 9000 0 0 –

2000 6000 3000 9000 9000 3000 1000 –

2000 6000 0 15,000 0 3000 1000 –

7000 9000 15,000 15,000 0 0 3000 –

9000 15,000 15,000 0 0 3000 4000 –

33,000 31,000 25,000 22,000 13,000 4000 1000 0

– – 120 110 80 50 – –

– – 40,000 37,000 13,000 4000 – –

Design and operation of heat exchangers and their networks

Table 6.7 Problem table for H2C2_150 (Zhu, 1997), Δtmin ¼ 10 K.

Optimal design of heat exchanger networks

263

180 160 QHU,min

140

t (°C)

120 100 Dtm QCU,min

80 60 Dtm 40

0

10,000

20,000 H (kW)

30,000

40,000

Fig. 6.6 Composite curves of Example 6.5 (H2C2_150) for Δtm ¼ 10 K.

H1

H2

E1

150

tpinch

CU1

E3

50

(200)

E2

170

tpinch

E4

40

(100)

HU1 120

tpinch–Dtm

CU2 50 (300)

tpinch–Dtm

110

80 (500)

C1 C2

HU2

Fig. 6.7 Network configuration of Example 6.5 (H2C2_150) (Zhu, 1997).

example, total annual cost (TAC), and a set of constraints describing the heat transfer and mass flow in the heat exchanger network. Generally, a mathematical programming method is carried out in three steps: At first, a network configuration including all possible network structures is set up. Then, a mathematic model is built, describing energy balance, mass balance and thermodynamic restrictions, and additional constraints for all possible heat exchangers and mixers in the network. Whether there is really a heat exchanger at a possible heat exchanger position is described by an integer

264

Design and operation of heat exchangers and their networks

variable. The heat exchanger area depends on the heat load or inlet and outlet temperatures of hot and cold streams; however, such a relation is not linear. Therefore, the mathematical model belongs to a mixed-integer nonlinear programming (MINLP) problem. The third step is to find out the solution of the MINLP model to reach the heat exchanger network featuring the optimal total annual cost and operability. Thus, the tasks of mathematical programming method include setting up the proper MINLP model and finding out its optimal solution with computer-based algorithms. Based on pinch technology, the first mathematical programming model is the transshipment model proposed by Papoulias and Grossmann (1983b) and Floudas et al. (1986). The transshipment model has the hot streams and heating utilities as commodity sources, the temperature intervals as intermediate nodes and cold streams, and cooling utilities as destinations. Heat is regarded as a commodity that is shipped from hot streams to cold streams through temperature intervals that account for thermodynamic constraints in the transfer of heat. By transshipment model, the search for the optimal network was decomposed into three major tasks. The first one involves the solution of a linear programming (LP) problem to target the process utility. In the next task, a mixed-integer linear programming (MILP) problem is solved to find the minimum number of matches needed to achieve maximum heat recovery (MHR target). To minimize the number of units, the authors normally apply the notion of process pinch to decompose the problem into two independent networks. By doing so, constrained utility targets implying a finite heat flow across the pinch can no longer be considered. Finally, a nonlinear nonconvex mathematical programming (NLP) model based on a network superstructure is tackled to search for the configuration featuring the lowest total area cost among those ones performing the set of heat matches and heat loads already selected through the MILP formulation, that is, the heat exchanger network at the level of structure. Yee and Grossmann (1990) proposed a MINLP mathematical formation where all the design decisions can be optimized simultaneously. The model is based on a superstructure resulting from a stagewise representation of heat exchanger networks where a match between any pair of hot and cold streams may take place at every stage. The number of stage is a model parameter to be adopted by the user in such a way that all possible network designs are taken into account. More stages will generally be required when the search is restricted to series configurations. A feature of the model is the linearity of the constraint set defining the problem feasible space. Such a linearity is achieved by assuming (a) isothermal mixing of streams, (b) no split stream

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265

flowing through two or more exchangers arranged in series, and (c) no stream bypass. Consequently, a constraint linearization scheme is not necessary, and a more reasonable computational time to solve the MINLP problem is required. However, the nonlinear, nonconvex objective function standing for the total annual cost of a heat exchanger network may still lead to local optimal solutions. Later, Daichendt and Grossmann (1994) proposed a preliminary screening procedure for the MINLP synthesis approach to lessen the computational difficulties originated by the large number of binary variables and nonconvexities. The procedure eliminates poor solutions from the original MINLP model by providing upper and lower bounds on the global optimum so as to render a reduced superstructure.

6.5 Stochastic and heuristic optimization algorithms Mathematical programming now is generally accepted by researchers for HENs due to the development of computer-based algorithm. When a MINLP model is established for synthesis of heat exchanger networks, the next step is to find a solving algorithm to search for the global optimal solution. For MINLP problems, the search/optimization methods can be classified in three main types: (a) calculus-based methods. These are subclassified into two groups: direct and indirect methods. Direct methods seek local optima by hopping on the objective function and moving in a direction related to the local gradient. Indirect methods, usually, solving a nonlinear set of equations using the objective function and its gradients, trying to seek the local extrema. The main drawbacks of these methods are the following: At first, they need analytical expressions of the objective function and its gradients. Also, these expressions must be differentiable. Second, in the case of multiple optima, these methods usually reach a local optimum without an indication whether the best optimum has been missed. In this case, it is necessary to restart the procedure to hopefully achieve further improvement. (b) Enumerative methods. These methods are fairly straightforward. They use a finite search space or a discretized infinite space and calculate the value of the objective function at every point. Of course, if the population of points is small, the task is not particularly difficult. However, the vast majority of problems consist of enormous number of points, and therefore, it is impossible for these methods to search each point individually. This is exactly the case for the synthesis problem dealing with large number of streams. (c) Random methods. These methods use random walks and

266

Design and operation of heat exchangers and their networks

random schemes and during the procedure save the best solutions. However, it is very rare to expect solutions better than solutions achieved with enumerative methods. (d) Randomized search methods. These methods use random choice as a tool to guide a highly exploitative search, coding first the parameter space. Efficient methods of this type are genetic algorithms and simulated annealing. The randomized search techniques have been used for identifying alternative near-optimal solutions or the optimum solution. The genetic algorithm, simulated annealing algorithm, particle swarm optimization algorithm, etc. can be applied for large-scale MINLP problem, but for complicated peaky problem, the final result also depends on both the initial value and the solving efficiency. For practical application, the efficient solving method is still highly demanded. On other hand, the development on thermal calculation of heat exchanger networks helps to lessen the search complexity of the network configurations. To avoid an arduous iterative calculation, Strelow (1984, 2000) proposed a general matrix method that can be applied to arbitrary two-stream heat exchanger networks. With a similar idea, Roetzel and Luo (2001) formulated a matrix algorithm for one-dimensional flow (cocurrent and countercurrent) multistream heat exchangers and their networks. Luo et al. (2002) extended this method to rating and sizing one-dimensional flow multistream heat exchangers and their networks based on a stagewise superstructure proposed by Yee et al. (1990). The advantage of using this method is obvious because the synthesis problem reduces into a general nonlinear optimization task with outlet temperature constraints and other additional constraints.

6.5.1 Genetic algorithm Genetic algorithms are randomized search algorithms that have been developed in an effort to imitate the mechanics of natural selection and natural genetics. Genetic algorithms operate on string structures, like biological structures, which are evolving in time according to the rule of survival of the fittest by using a randomized yet structured information exchange. Thus, in every generation, a new set of strings is created, using parts of the fittest members of the old set. The main characteristics of a genetic algorithm are as follows: (1) The genetic algorithm works with a coding of the parameter set, not the parameters themselves. (2) The genetic algorithm initiates its search from a population of points, not a single point.

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267

(3) The genetic algorithm uses payoff information, not derivatives. (4) The genetic algorithm uses probabilistic transition rules, not deterministic ones. At first, the coding to be used must be defined. Then using a random process, an initial population of strings is created. Next, a set of operators is used to take this initial population to generate successive populations, which hopefully improve with time. The main operators of the genetic algorithms are reproduction, crossover, and mutation. Reproduction is a process based on the objective function (fitness function) of each string. This objective function identifies how “good” a string is. Thus, strings with higher fitness value have bigger probability of contributing offsprings to the next generation. Crossover is a process in which members of the last population are mated at random in the mating pool. So, a pair of offsprings is generated, combining elements from two parents (members), which hopefully have improved fitness values. Mutation is the occasional (with small probability) random alteration of the value of a string position. In fact, mutation is a process of random walk through the coded parameter space. Its purpose is to ensure that important information contained within strings may not be lost prematurely. The implementation of GAs to the problem of optimization of operation and site-scale energy use in production plants is envisaged to be carried along the following lines. First, using the pinch design method, the problem requirements and constraints will be defined. Next, the region of the pinch will be identified, and the essential matches will be made. Then, the problem will be coded by generating the appropriate strings. These strings will contain the general features and the parameters, which affect the problem. Each string will represent a possible network configuration. For each one of the strings, the objective function will be calculated, that is, the value of energy and utility usage and the number of units need to be used in the represented network configuration, that is, the total capital and operating cost. Initially, a starting population of strings will be created using a random procedure. The three main operators of the genetic algorithms will be performed to improve the value of the objective function, namely, to create network configurations with minimal total capital and operating cost. To the final population of strings/possible networks, advanced techniques will be applied for further improvement.

268

Design and operation of heat exchangers and their networks

Genetic algorithm is a kind of stochastic algorithm based on the theory of probability. In application this method to a stagewise superstructure model, the search process is determined by stochastic strategy. The global optimal solution for the synthesis of heat exchanger networks can be obtained at certain probability. The search process begins with a set of initial stochastic solutions, which is called “population.” Each solution is called “chromosome,” the chromosome is composed of “gene,” and the “gene” stands for the optimal variables of heat exchanger networks, for example, the mass flowrates of cold streams and hot streams. There are two kinds of calculation operation in the genetic algorithm: genetic operation and evolution operation. The genetic operation adopts the transferring principle of probability, selects some good chromosomes to propagate at certain probability, and lets the other inferior chromosomes to die; thus, the search direction will be guided to the most promising region. With a stochastic search technique, they can explore different regions of the search space simultaneously and hence are less prone to terminate in local minimum. The strength of the genetic algorithm is the exploration of different regions of the search space in relatively short computation time. Furthermore, multiple and complex objectives can easily be included. But genetic algorithm provides only a general framework for solving complex optimization problem. The genetic operators are often problem-dependent and are of critical importance for successful use in practical problem. Specifically, to the synthesis problem of heat exchanger networks with multistream heat exchangers, an approach for initial network generation, heat load determination of a match within superstructure should be given. Some operators such as crossover operator, mutation operator, orthogonal crossover, and effective crowding operators are appropriately designed to adapt to the synthesis problem. Another difficulty for genetic algorithm application is the treatment of constraints. During the genetic evolution, an individual of the population may turn into infeasible solution after manipulated by genetic operators, which will lead to failure to find a feasible solution during evolution, especially for the optimization problem with strict constraints. Hence, some strategy should be contrived for constraints guarantee in genetic computation.

6.5.2 Simulated annealing algorithm Another effective algorithm used to solve large-scale combinatorial optimization problems is the simulated annealing algorithm (Kirkpatrick et al., 1983).

Optimal design of heat exchanger networks

269

Similar to the genetic algorithm, it is also in principle a random method and generally can handle discontinuous, nondifferential, and nonconvex function. Unlike the genetic algorithm, the convergence property of the simulated annealing algorithm can be proved theoretically, and the accepted points are in Boltzmann distribution at a constant temperature. The genetic algorithm begins from many initial points and is inherently parallel, while the simulated annealing algorithm starts from a single point. By using the Metropolis rule to accept the worst solution in a fraction, the fraction gradually approaches to zero. It is possible to jump out of a local optimum to search for the global optimum. The solution points in the simulated annealing algorithm satisfy Boltzmann distribution.

6.5.3 Particle swarm optimization algorithm The particle swarm optimization is a heuristic algorithm originally proposed by Kennedy and Eberhart (1995). It works like the movement of a bird flock in which the individual birds are guided by their own experience and the experience of the neighboring birds. In the particle swarm optimization, the bird flock is represented as a population (called a swarm) of candidate solutions (called particles). If improved positions are being discovered by one or some individual particles, the movement of the swarm will be attracted and move to these positions in their own ways, and during the movement, their own experience will be updated. Similar to the genetic algorithm, the particle swarm optimization is less sensitive to the starting point of a solution, and due to the stochastic velocity and acceleration of the particle movements, the local optimum traps can be avoided. Silva et al. (2008, 2010) applied the particle swarm optimization approach to the synthesis of heat exchanger networks. Huo et al. (2013) presented a two-level approach, in which the operators of the genetic algorithm handled the structure optimization, while the lower level handled the continuous variables with a standard particle swarm optimization algorithm. Although the standard particle swarm optimization algorithm is capable of detecting the region of attraction, it cannot perform a refined local search to find the optimum with high accuracy. In fact, if a particle has learned from its neighboring particles and has flied to the region of attraction, it should be able to make an own search for the best position. Wang et al. (2017) presented a comprehensive simultaneous synthesis approach based on stagewise superstructure to design cost-optimal heat exchanger network. They employed a two-level optimization algorithm for solving the synthesis

270

Design and operation of heat exchangers and their networks

problems. In the master-level optimization, an evolutionary method was developed for generating network configurations that are then sent to the slave level for continuous variable optimization using a memetic particle swarm optimization algorithm. Their case studies showed an excellent search ability of the bilevel algorithm.

6.5.4 Knowledge-based expert system Recently, there has been a very rapid growth in the application of knowledge-based system in synthesis of heat exchanger networks. For a given network design problem, there are large numbers of possible configurations, and the optimal solution cannot be obtained by a simple search method. An efficient method is to establish a mathematical model of network superstructure. For the available superstructures, there is possibility for each hot stream and each cold stream to match; the superstructure will be huge in scale if number of hot and cold streams exceeds a certain number, which makes the search for optimal solution complicated and difficult to be obtained, especially when the number of streams is large. Furthermore, in the whole process of simultaneous synthesis of heat exchanger networks by adopting stagewise superstructure, the network configuration was kept as a black box, and experienced experts were kept outdoor in the process and cannot help with the scheme choice. In addition, the giant search scope makes the calculation a complete and arduous job. While from the viewpoint of engineering application, some of hot stream and cold stream matches involved in the superstructure are not necessary; on the other hand, not every possible structure is contained in those superstructures. An expert system can help to present a more reasonable and relatively simple superstructure to make search for the optimal solution easy to realize. The expert system method is an artificial intelligence technique. The principles in expert system method are a set of rules based on logical knowledge and experiences such as whether the stream should be split or not, how to determine of split number if the split of a stream is necessary, and the match principle of streams. These principles are translated into computer language in the calculation model. Generally, an expert system consists of a database, a knowledge base, control strategies, and a man-machine interface. The database is divided into two parts: static and dynamic bases. The static base records the initially assigned data and relevant data required for solving the problem, and the dynamic base stores all the intermediate information generated while solving

Optimal design of heat exchanger networks

271

the problem and the final results. In the data base, each stream is represented by a list. For example, a hot stream is characterized by a list of the number of hot stream i, its input temperature and output temperatures and the heatcapacity flowrate. Therefore, a set of hot or cold streams is defined by a hot or cold list, labeled as H or C. All forbidden matches between hot stream i and cold stream j are specified by a list named R. Knowledge base in the expert system must be explicitly represented. The function of this part is to produce rules that represent behavioral knowledge in an “if-then” form consisting of a conditional part and an action part. The third-part controlling strategies are used to manage and apply the available information stored in the knowledge base to change the problem state in the dynamic database from an initial state to the solution state through a sequence of intermediate states. The knowledge rules are obtained from engineering experiences. For example, some of the rules are illustrated as follows: (1) If the input temperature of hot stream i is less than or equal to the input temperature of cold stream j, the match between hot stream i and cold stream j is forbidden. (2) If the corrosiveness of hot stream i or cold stream j is serious, the match between hot stream i and cold stream j is forbidden. (3) If the distance between hot stream i and cold stream j is long enough, then the match between those two streams is forbidden. In addition, the number of HEN stages can be also decided with the expert system. And after the expert system is developed, the rule bases can be also replenished and perfected. From earlier, it can be seen that the expert system controls the interaction between the various problem aspects and generates the models to be solved. The apparent advantage of using an expert system is the flexibility in applying or to combine several methods for subproblems. It is easy to modify or improve the capabilities of the system by expanding the contents of the knowledge base.

6.6 Examples of heat exchanger network synthesis Because of the high complexity of the synthesis problems for optimal design of heat exchanger networks, there is no available methods to prove whether a network has reached its global minimum of the total annual cost (TAC). A common way to check a newly developed synthesis procedure is carry out several case studies using available examples in the literature, which have already been investigated and optimized by other researchers with other

272

Design and operation of heat exchangers and their networks

procedures. Such examples are summarized in the succeeding text with their up-to-date global minimum of TAC and the corresponding structures. We will label such examples with the number of hot streams and cold streams. Each example is revised according to the given optimal network configuration and original problem data, calculated with the exact equation of logarithmic mean temperature difference, and optimized to its local minimum of TAC using a local optimizing strategy (Luo et al., 2009). Example H1C2 This example was used by Biegler et al. (1997, Table 16.6). The problem data are listed in Table 6.8. The best network was obtained by Huang et al. (2012) based on the stagewise superstructure with nonisothermal mixing in the networks and is shown in Fig. 6.8. This network has two independent variables that can be further optimized. Such variables are shown in the figure with more decimal places than others. Table 6.8 Problem data for H1C2 (Biegler et al., 1997). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 C1 C2 HU CU

167 76 47 227 27

77 157 95 227 47

22 20 7.5

2 2 0.67 1 1

110 10

Heat exchanger cost ¼ 1200A0.57$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Huang et al. (2012), Huang and Karimi (2013) Biegler et al. (1997)

76,327 77,972

76,354 77,913

H1

167

1620

258.4

(20.66325)

77

(22)

76

157

(20)

47

95 101.5 686

(7.5)

C1 C2

Fig. 6.8 Optimal solution for Example H1C2 (Huang et al., 2012), TAC ¼ 76,354$/yr.

Optimal design of heat exchanger networks

273

Example H2C1 This example was used by Shenoy et al. (1998) as a synthesis problem with multiple utilities. The problem data are listed in Table 6.9. The best network was obtained by Na et al. (2015). For solving such problems, they developed a modified superstructure that contains a utility substage for use in considering multiple utilities in a simultaneous MINLP model. This network has eight independent variables. Optimizing these variables yields the minimum TAC of 96,041$/yr, as is shown in Fig. 6.9. Table 6.9 Problem data for H2C1 (Shenoy et al., 1998). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 HU1 HU2 HU3 CU

105 185 25 210 160 130 5

105 185 25 210 160 130 5

10 5 7.5

0.5 0.5 0.5 5 5 5 2.6

160 110 50 10

Heat exchanger cost ¼ 0.298  800A$/yr (A in m2) (Plant lifetime: 5 years; loan rate of interest: 15%; annualization factor: 0.298) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Na et al. (2015) Shenoy et al. (1998) Ponce-Ortega et al. (2010)a

96,076 96,412 97,079

96,041 96,592 97,043

a

No stream split.

H1 H2

281.6534

105

518.3 25

(10)

97.11261

185 (5)

199.3 185

150.0322

55.91801

152.7995

35

122.1138

209.0

(7.5)

HU1

HU2

HU3

25

C1

(2.503083)

141.0872

Fig. 6.9 Optimal solution for Example H2C1 (Na et al., 2015), TAC ¼ 96,041$/yr.

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Design and operation of heat exchangers and their networks

Example H2C2_150 This example was used by Zhu (1997) who took the stream data from Linnhoff and Ahmad (1990, Fig. 6) and cost data from Ahmad et al. (1990). The problem data are listed in Table 6.10. The best solution without stream split was found by Pava˜o et al. (2016), Zhang et al. (2016a,b), and Wang et al. (2017), with the revised TAC of 1,809,487$/yr. We applied the hybrid genetic algorithm based on the stagewise superstructure (Luo et al., 2009) to solve this problem. As is shown in Fig. 6.10, the obtained network has one stream split and four independent variables, and the minimal TAC is 1,805,242$/yr. Table 6.10 Problem data for H2C2_150 (Zhu, 1997). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 HU CU

150 170 50 80 180 20

50 40 120 110 180 40

200 100 300 500

0.2 0.2 0.2 0.2 0.2 0.2

110 10

Heat exchanger cost ¼ 0.295260  (30,800 + 750A0.81) $/yr (A in m2) (Plant lifetime: 6 years; rate of interest: 10%; annualization factor: 0.29526) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Own work Pava˜o et al. (2016)a Zhang et al. (2016a,b)a Wang et al. (2017)a Zhu (1997) Silva et al. (2010)

– 1,814,000 1,807,805 1,809,487 1,550,000 1,816,470

1,805,242 1,809,487

a

1,815,294

No stream split.

H1 H2

18442. 362

150

1558 50

(200)

8354. 627

170

1573. 831

40

(100)

984

3072

120

50 (300)

(51.27071)

110

80 (500)

C1 C2

6645

Fig. 6.10 Optimal solution for Example H2C2_150, TAC ¼ 1,805,242$/yr.

Optimal design of heat exchanger networks

275

Example H2C2_175 This example was taken by Isafiade and Fraser (2008) from the literature. The problem data are listed in Table 6.11. The best network configuration published in the literature was proposed by Shenoy (1995) (see Fig. 2 of Isafiade and Fraser, 2008) and Azeez et al. (2013). This network contains four independent variables that can be further optimized, yielding the local minimal TAC of 234,285$/yr. Using the hybrid genetic algorithm based on the stagewise superstructure (Luo et al., 2009), we obtained the best network shown in Fig. 6.11, with TAC ¼ 228,546$/yr. Table 6.11 Problem data for H2C2_175 (Isafiade and Fraser, 2008). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 HU CU

175 125 20 40 180 15

45 65 155 112 179 25

10 40 20 15

0.2 0.2 0.2 0.2 0.2 0.2

120 10

Heat exchanger cost ¼ 0.322  (30,000 + 750A0.81) $/yr (A in m2) (Plant lifetime: 5 years; rate of interest: 10%; annualization factor: 0.322) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Own work Azeez et al. (2013) Isafiade and Fraser (2008)

– 235,931 237,800

228,546 234,285 236,363

H1

1300

175

45

(10)

313.0 H2

1006. 962

125

(18.16548)

65

(40)

393.0 155

20 (20)

112

(10.37170)

40 1080

(15)

C1

C2

Fig. 6.11 Optimal solution for Example H2C2_175, TAC ¼ 228,546$/yr.

276

Design and operation of heat exchangers and their networks

Example H2C2_260 This example was originally introduced by Ahmad (1985, p. 315, Fig. A2.15), in which the heat transfer coefficients for all matches were 1.5 kW/m2 K. Nielsen et al. (1996) used the data for the network design considering the minimum number of 1–2 shells in an exchanger based on a specified effectiveness parameter Xp ¼ 0.9. The heat transfer coefficients for all matches were 0.4 kW/m2 K. Khorasany and Fesanghary (2009) took these data for the network design with common counterflow heat exchangers. According to the problem data given in Table 6.12, the best network configuration was found first by Huo et al. (2012) with TAC ¼ 11,632$/yr and later by Myankooh and Shafiei (2015) with the same configuration and three optimized variables, as is shown in Fig. 6.12, which yields the minimal TAC of 11,540$/yr. Table 6.12 Problem data for H2C2_260 (Khorasany and Fesanghary, 2009). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 HU CU

260 250 120 180 280 30

160 130 235 240 279 80

3 1.5 2 4

0.4 0.4 0.4 0.4 0.4 0.4

110 12.2

Heat exchanger cost ¼ 300A0.5 $/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Huo et al. (2012)a Myankooh and Shafiei (2015, 2016)a Khorasany and Fesanghary (2009)a

11,632 11,540 11,895

11,540

a

11,802

No stream split.

H1 H2

234.3 191

260

65.68

160

(3)

28.07 111.7 409

250

40.18 850

130

(1.5)

12.39 235

120 (2)

240

180 (4)

C1 C2

5.681

Fig. 6.12 Optimal solution for Example H2C2_260 (Huo et al., 2012; Myankooh and Shafiei, 2015, 2016), TAC ¼ 11,540$/yr.

Optimal design of heat exchanger networks

277

Example H2C2_270 This example was originally introduced by Gundersen (2002). The problem data were given by Escobar and Trierweiler (2013) and are listed in Table 6.13. They used the General Algebraic Modeling System (GAMS) to solve this problem. However, their solutions did not approach to the local minimal TAC of the corresponding networks. After the optimization of the independent variables, in their network shown in Fig. A2(b) of their paper, the bypass of H1 after the exchanger H1C2 is eliminated, and the TAC of the network approaches to 351,411$/yr. Later, the better solution was found by Stegner et al. (2014) by the use of an enhanced vertical heat exchange algorithm, in which a new form of graphic depiction of the problem’s data in a temperature-enthalpy diagram was implemented. The network is shown in Fig. 6.13, which contains only one independent variable, and the revised TAC is 350,108$/yr. Table 6.13 Problem data for H2C2_270 (Escobar and Trierweiler, 2013). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 HU CU

270 220 50 160 250 15

160 60 210 210 250 20

18 22 20 50

0.5 0.5 0.5 0.5 1.5 1

200 20

Heat exchanger cost ¼ 4000 + 500A0.83$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Stegner et al. (2014)a Escobar and Trierweiler (2013)

349,316 361,983

350,108 351,411

a

No stream split.

H1

H2

71.40

1908. 598

270

160

(18)

3200

220

60

(22)

320 50

210

(20)

C1

591.4 210

160 (50)

C2

Fig. 6.13 Optimal solution for Example H2C2_270 (Stegner et al., 2014), TAC ¼ 350,108$/yr.

278

Design and operation of heat exchangers and their networks

Example H2C2_300 This example is taken from Ahmad (1985, p.306, Fig. A2.1). The problem data are listed in Table 6.14. The network obtained by Ahmad has the TAC of 7421.5$/yr. Using particle swarm optimization, Silva et al. (2010) obtained their optimal solution, of which the reported TAC is 7884$/yr (the revised value should be 8830$/yr according to the parameters given in Fig. 4 of their paper). Their network contains six independent variables. However, by optimizing these variables, two of them, QHUC1 and QH1C1, approach to zero, that is, the heater for stream C1 and the exchanger for streams H1 and C1 can be removed, yielding the minimal TAC of 7408$/yr. The final design of the network is shown in Fig. 6.14. Table 6.14 Problem data for H2C2_300 (Ahmad, 1985, p.306, Fig. A2.1). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 HU CU

300 200 40 140 400 10

80 40 180 280 399 11

0.3 0.45 0.4 0.6

0.4 0.4 0.4 0.4 0.4 0.4

110 12.2

Heat exchanger cost ¼ 300A0.5 $/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Silva et al. (2010) Ahmad (1985, p.314, Table A2.2 A2C2)a

7884 7421

7408 7421.5

a

No stream split.

19.29 H1 H2

46.71 466

300

80

(0.3)

4.111 555

200

40

(0.45)

11.89

180 56

33.17

40 (0.4)

140

280

(0.6)

C1 C2

(0.1028495)

Fig. 6.14 Optimal solution for Example H2C2_300 (Silva et al., 2010), TAC ¼ 7408$/yr.

Optimal design of heat exchanger networks

279

Example H2C2_320F This example is 4SP1 problem taken from Lee et al. (1970). Papoulias and Grossmann (1983a) and Bagajewicz et al. (1998) transferred the problem data from English units to international system of units (SI). For the unique revision and comparison of the local minimal TAC, we would like to take the problem data in its original English units, which were explicit given by Yerramsetty and Murty (2008) and are listed in Table 6.15. The best network configuration was found by Azeez et al. (2012). After optimizing the two independent variables, the TAC of this network approaches to the minimum of 10,582$/yr, as is shown in Fig. 6.15. Table 6.15 Problem data for H2C2_320F (Yerramsetty and Murty, 2008). Stream

Tin (°F)

Tout (°F)

C_ (kBtu/h°F)

α (kBtu/ft2°F)

Cost ($/kBtu yr)

H1 H2 C1 C2 HU CU

320 480 140 240 540 100

200 280 320 500 540 180

16.6668 20 14.4501 11.53

0.3 0.3 0.3 0.3 0.6 0.3

12.76 5.24

Heat exchanger cost ¼ 35A0.6 $/yr (A in ft2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Azeez et al. (2012) Nishida et al. (1977)a Papoulias and Grossmann (1983a)a,b Bagajewicz et al. (1998)a,b Yerramsetty and Murty (2008)a Azeez et al. (2013) Pho and Lapidus (1973)a Lee et al. (1970)a

10,786 13,590 – 10,580 10,782 10,794 13,685 13,481

10,582 10,587

a

10,615 10,631 13,670

No stream split. Using SI units.

b

1353.4 H1 H2

320

(11.77133)

200

(16.6668)

2752. 428

480 (20)

320

140 245.4

500

280 646.6

1247.6

(14.4501)

240 (11.53)

C1 C2

Fig. 6.15 Optimal solution for Example H2C2_320F (Azeez et al., 2012), TAC ¼ 10,582$/yr.

280

Design and operation of heat exchangers and their networks

Example H2C2_443K This example is taken from Linnhoff et al. (1982) by Yee and Grossmann (1990), as is listed in Table 6.16. The best network without stream split was found by Wang et al. (2017), of which TAC¼ 79,233$/yr. If the stream split is allowed, we can find even better results shown in Fig. 6.16 with TAC ¼ 77,964$/yr, using our hybrid genetic algorithm (Luo et al., 2009). Table 6.16 Problem data for H2C2_443K (Yee and Grossmann, 1990). Stream

Tin (K)

Tout (K)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 HU CU

443 423 293 353 450 293

333 303 408 413 450 313

30 15 20 40

1.6 1.6 1.6 1.6 4.8 1.6

80 20

Heat exchanger cost (except heaters) ¼ 1000A0.6 $/yr (A in m2) Heat exchanger cost for heaters ¼ 1200A0.6 $/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Own work Wang et al. (2017)a Yee and Grossmann (1990) Zhu et al. (1995) Yee and Grossmann (1990)a Azeez et al. (2013)

– 79,233 80,274 80,815 89,832 90,521

77,964 79,233 80,272 80,291 85,353 87,594

a

No stream split.

H1 H2

443

201.7 2400 298.3 353

400 293

(30)

1800

423

353

(15)

408

333 (20)

413

(15.10036)

303 (40)

C1 C2

Fig. 6.16 Optimal solution for Example H2C2_443K, TAC ¼ 77,964$/yr.

Optimal design of heat exchanger networks

281

Example H2C2_443KZ This example is a variant of Example H2C2_443K with different cost parameters used by Zamora and Grossmann (1998). The problem data are given in Table 6.17. Toffolo (2009) developed a two-level optimization algorithm, in which the network configuration is generated with an evolutionary algorithm by applying genetic operators to a graphbased representation of the network. With this method, they found the best network differing from the stagewise superstructure, as is shown in Fig. 6.17. The network contains two splits and has three independent variables, and its TAC is 82,363$/yr. Table 6.17 Problem data for H2C2_443KZ (Zamora and Grossmann, 1998). Stream

Tin (K)

Tout (K)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 HU CU

443 423 293 353 450 293

333 303 408 413 450 313

30 15 20 40

1.6 1.6 1.6 1.6 4.8 1.6

80 20

Heat exchanger cost (except heaters) ¼ 6250 + 83.26A$/yr (A in m2) Heat exchanger cost for heaters ¼ 6250 + 99.91A$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Toffolo (2009) Pariyani et al. (2006, Fig. 2(b)) Pettersson (2008) Yerramsetty and Murty (2008) Zamora and Grossmann (1998)a Adjiman et al. (2000)a Pariyani et al. (2006, Fig. 2(a))a Gorji-Bandpy et al. (2011)

82,363 85,307 84,066 84,222 85,968 74,711 85,972 67,174

82,363 83,937

a

85,970

Infeasible

No stream split.

270.1 041 H1 H2

2400 629.9

443

(27.59909)

400 293

(30)

423

353

(15)

408

333 (20)

413

(17.23082)

1400

303 (40)

C1 C2

Fig. 6.17 Optimal solution for Example H2C2_443KZ (Toffolo, 2009), TAC¼ 82,363$/yr.

282

Design and operation of heat exchangers and their networks

Example H2C3 This example is used by Shenoy et al. (1998) for multiple utilities targeting of heat exchanger networks. The problem data are given in Table 6.18. The best network was obtained by Huang and Karimi (2013) with a simultaneous synthesis approach based on the stagewise hyperstructure. The network is shown in Fig. 6.18, which consists of two subnetworks {H1, C2, C3} and {H2, C1}. The network has one split and four independent variables, and the revised TAC of the network is 1,116,629$/yr. Table 6.18 Problem data for H2C3 (Shenoy et al., 1998). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 C3 HU1 HU2 HU3 CU1 CU2

155 230 115 50 60 255 205 150 30 40

85 40 210 180 175 254 204 149 40 65

150 85 140 55 60

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

70 50 20 10 5

Heat exchanger cost ¼ 0.322  (13,000 + 1000A0.83) $/yr (A in m2) (Plant lifetime: 5 years; rate of interest: 10%; annualization factor: 0.322) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Huang and Karimi (2013) Ponce-Ortega et al. (2010) Huang and Karimi (2014) Na et al. (2015) Shenoy et al. (1998) Isafiade and Fraser (2008)

1,115,868 1,121,175 1,120,271 1,120,609 1,158,500 1,150,460

1,116,630 1,116,787

H1 H2

5066. 436

155

(72.26217)

1,117,381 1,121,698

CU1 512

(150)

85 7047

230

40

(85)

4197 210 HU1 180 HU2

2084

CU1 9102. 914

115 (140)

50 (55)

60

175 HU2 1979

4921. 479

(60)

C1 C2 C3

Fig. 6.18 Optimal solution for Example H2C3 (Huang and Karimi, 2013), TAC ¼ 1,116,630$/yr.

Optimal design of heat exchanger networks

283

Example H3C2 This example is taken from Bjork and Westerlund (2002). The problem data are listed in Table 6.19. The best network was obtained by Huang and Karimi (2013) with the simultaneous synthesis approach based on the stagewise hyperstructure. The network is shown in Fig. 6.19, which contains one split and one bypass between the splitted streams. The revised TAC of the network is 94,751$/yr. Table 6.19 Problem data for H3C2 (Bjork and Westerlund, 2002). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 C1 C2 HU CU

155 80 200 20 20 220 20

30 40 40 160 100 220 30

8 15 15 20 15

2 2 2 2 2 2 2

120 20

Heat exchanger cost ¼ 6000 + 600A0.85$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Huang and Karimi (2013) Bjork and Westerlund (2002) Huang et al. (2012) Pava˜o et al. (2016) Bjork and Westerlund (2002) Laukkanen et al. (2012)

94,742 96,001 95,643 95,660 100,720 –

94,751 95,660

H1 H2 H3

1000

155

100,691

30

(8)

600

80

40

(15)

200

1200

1200

40

(15)

20

160 (1.366119)

100

(11.27517)

(20)

20 (15)

C1 C2

Fig. 6.19 Optimal solution for Example H3C2 (Huang and Karimi, 2013), TAC ¼ 94,751$/yr.

284

Design and operation of heat exchangers and their networks

Example H2C4 This example is also taken from Bjork and Westerlund (2002). The problem data are listed in Table 6.20. Their reported TAC was 139,083$/yr; however, the network was not given. Till now, the best network was obtained by Huang and Karimi (2012, 2013) with the simultaneous synthesis approach based on the stagewise hyperstructure. The network is shown in Fig. 6.20, which contains three splits and one bypass. There are six independent variables, which can be optimized. The revised TAC of the network is 122,940$/yr. Table 6.20 Problem data for H2C4 (Bjork and Westerlund, 2002). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 C1 C2 C3 C4 HU CU

180 240 40 120 40 80 325 25

75 60 230 260 130 190 325 40

30 40 20 15 25 20

2 2 1.5 1.5 2 2 1 2

120 20

Heat exchanger cost ¼ 8000 + 50A0.75$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Huang and Karimi (2012, 2013) Bjork and Westerlund (2002) Bergamini et al. (2007)

123,398 139,083 140,367

122,940 – 131,692

H1

22.65

180

2250

(29.51761)

H2

75

(30)

3777 (20.03153)

240

60

(40) (21.16799)

40

230 305 260

305

(19.83485)

120 (15)

877.3 453 130

(20)

917.8 743

40 (25)

190

80 2200

(20)

C1 C2 C3 C4

Fig. 6.20 Optimal solution for Example H2C4 (Huang and Karimi, 2012, 2013), TAC ¼ 122,940$/yr.

Optimal design of heat exchanger networks

285

Example H3C3 This example is taken from Lee et al. (1970) and known as 6SP1 problem. The problem data are listed in Table 6.21. Because the utility costs have not been explicitly given in the literature, they are calculated here by the use of RefProp for the latent heat (Δhfg ¼ 768.33 Btu/lbm at ps ¼ 450 psia). The equipment downtime is 380 hours per year, which yields the hot utility cost of 10.906$/kBtu/yr and the cold utility cost of 5.2375$/kBtu/yr, respectively. The best network was found by Nishida et al. (1977), which has no independent variable. Therefore, there is a unique TAC of 35,010$/yr, as is reported by Nishida et al. In fact, the network obtained by Lee et al. (1970) was already close to this best network. After the local optimization, we found that the unit H2C2 in the network of Lee et al. vanished, resulting to the best network shown in Fig. 6.21. Table 6.21 Problem data for H3C3 (Lee et al., 1970). Stream

Tin (°F)

Tout (°F)

C_ (kBtu/h°F)

α (kBtu/ft2°F)

Cost ($/kBtu yr)

H1 H2 H3 C1 C2 C3 HU CU

440 520 390 100 180 200 456 100

150 300 150 430 350 400 456 180

28 23.8 33.6 16 32.76 26.35

0.3 0.3 0.3 0.3 0.3 0.3 0.6 0.3

10.906 5.2375

Heat exchanger cost ¼ 35A0.6 $/yr (A in ft2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Lee et al. (1970)a Nishida et al. (1977)a Chakraborty and Ghosh (1999)a Pho and Lapidus (1973)a

35,108b 35,010 35,016 35,659c

35,010

35,526

a

No stream split. After revision, H2C2 is deleted. c After revision, H2CU is deleted. b

Continued

286

Design and operation of heat exchangers and their networks

Example H3C3—cont’d 2806 H1 H2 H3

440 5270

44

150

(28)

520

5236

300

(23.8)

5569.2

390

2494.8 150

(33.6)

100

430

(16)

180

350

(32.76)

200

400

(26.35)

C1 C2 C3

Fig. 6.21 Optimal solution for Example H3C3 (Nishida et al., 1977), TAC ¼ 35,010$/yr.

Example H5C1 This example was first investigated by Yee and Grossmann (1990) and later by many other researchers. The problem data are listed in Table 6.22. The best network was obtained by Huang and Karimi (2013) with the simultaneous synthesis approach based on the stagewise hyperstructure, as is shown in Fig. 6.22, containing two splits and one bypass. There are four independent variables. The optimization of the four variables yields the minimal TAC of 570,391$/yr. Table 6.22 Problem data for H5C1 (Yee and Grossmann, 1990). Stream

Tin (K)

Tout (K)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 H5 C1 HU CU

500 480 460 380 380 290 700 300

320 380 360 360 320 660 700 320

6 4 6 20 12 18

2 2 2 2 2 2 2 2

140 10

Heat exchanger cost ¼ 1200A0.6$/yr (A in m2)

Optimal design of heat exchanger networks

287

Example H5C1—cont’d Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Huang and Karimi (2013) Toffolo (2009) Luo et al. (2009) Huang et al. (2012) Pava˜o et al. (2016) Lewin (1998) Khorasany and Fesanghary (2009) Wei et al. (2004b) Luo et al. (2004) for multistream HE Yee and Grossmann (1990) Isafiade and Fraser (2008) Azeez et al. (2013)

570,362 570,900 571,698 571,657 571,737 573,205 572,476 571,585 – 575,595 581,900 580,023a

570,391 570,764 571,698

a

572,152 572,734 573,135 573,807 576,625

After revision, one H1C1 is deleted.

H1 H2 H3 H4 H5

1080

500

320

(6)

400

480

380

(4)

110

489.8 918

460

360

(6)

400

380

360

(20)

720

380

320

(12)

3570 290

660 (11.82453)

(18)

C1

(14.47765)

(7.054784)

Fig. 6.22 Optimal solution for Example H5C1 (Huang and Karimi, 2013), TAC ¼ 570,391$/yr.

288

Design and operation of heat exchangers and their networks

Example H3C4 This example was originally used by Colberg and Morari (1990). The unit costs and utility costs were supplemented by Yee and Grossmann (1990) and Xiao et al. (2006), respectively (see Table 6.23). The best network without stream split was found by Wang et al. (2017), of which TAC ¼ 79,233$/yr. Using the hybrid genetic algorithm based on the stagewise superstructure (Luo et al., 2009), we find the better solution shown in Fig. 6.23. The network consists of two subnetworks {H1, H2, C1, C2} and {H3, C3, C4}. Each subnetwork has an independent variable. The local optimization yields TAC ¼ 176,200$/yr.

Table 6.23 Problem data for H3C4 (Xiao et al., 2006). Stream

Tin (K)

Tout (K)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 C1 C2 C3 C4 HU CU

626 620 528 497 389 326 313 650 293

586 519 353 613 576 386 566 650 308

9.802 2.931 6.161 7.179 0.641 7.627 1.69

1.25 0.05 3.2 0.65 0.25 0.33 3.2 3.5 3.5

130 20

Heat exchanger cost ¼ 8600 + 670A0.83$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Own work Wang et al. (2017)a Xiao et al. (2006) for multistream HE Yee and Grossmann (1990) Isafiade and Fraser (2008)

– 183,029 – – 168,700

176,200 183,029 183,332 185,106 188,002

a

No stream split.

Optimal design of heat exchanger networks

289

Example H3C4—cont’d H1 H2

392

626

586

(9.802)

176

620

519

(1.991076)

(2.931)

261 H3

528

353

(6.161)

265 497

613

(7.179)

389

576

(0.641)

120

326

386 458

68 566

(7.627)

313 359.1 642

C1 C2 C3 C4

(1.69)

Fig. 6.23 Optimal solution for Example H3C4, TAC ¼ 176,200$/yr.

Example H4C3 This example was used by Ciric and Floudas (1991) for illustrating their Strict-pinch transshipment model. The problem data are given in Table 6.24. The best network was obtained by Huang and Karimi (2012, 2013) based on the stagewise hyperstructure, as is shown in Fig. 6.24, with three independent variables and TAC ¼ 105,426$/yr. Table 6.24 Problem data for H4C3 (Ciric and Floudas, 1991). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 C1 C2 C3 HU CU

160 249 227 271 96 116 140 300 70

110 138 106 146 160 217 250 300 90

7.032 8.44 11.816 7 9.144 7.296 18

1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6

80 20

Heat exchanger cost ¼ 1300A0.6 $/yr (A in m2) Continued

290

Design and operation of heat exchangers and their networks

Example H4C3—cont’d Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Huang and Karimi (2012, 2013) Ciric and Floudas (1991, Fig. 7) Vidyashankar and Narasimhan (2008) Xiao et al. (1993) Lv et al. (2017)

105,403 114,460a 108,838 109,195 109,181

105,426 106,877 108,047 108,575

a

After revision, H1C3, H2C3, and H4C2 are deleted.

351.6 H1 H2 H3 H4

160

110

(7.032)

736.9

199.9

249

138

(8.44)

149.9 585.2

694.6 384

227

106

(11.816)

875

271

146

(7)

160

96 (9.144)

116

217

(7.296)

C1 C2

210.4 250

140 (18)

C3

(8.191837)

Fig. 6.24 Optimal solution for Example H4C3, TAC ¼ 105,426$/yr (Huang and Karimi, 2012).

Example H6C1 This example was taken from Papoulias and Grossmann (1983a). The overall heat transfer coefficients can be found in the literature (Ciric and Floudas, 1991). For the unique revision and comparison of the local minimal TACT, we keep the problem data in its original English units and supplement the heat transfer coefficient of H5 to be 0.06 kBtu/ft2°F, as is shown in Table 6.25. The best network was obtained by Ciric and Floudas (1991) shown in Fig. 6.25, with five independent variables and the revised TAC ¼ 639,188$/yr.

Optimal design of heat exchanger networks

291

Example H6C1—cont’d Table 6.25 Problem data for H6C1 (Ciric and Floudas, 1991). Stream

Tin (°F)

Tout (°F)

C_ (kBtu/h°F)

α (kBtu/ft2°F)a

H1 H2 H3 H4 H5 H6 C1 HU CU

675 590 540 430 400 300 60 800 80

150 450 115 345 100 230 710 800 140

15 11 4.5 60 12 125 47

0.24 0.118309859 0.092307692 0.118309859 0.06 0.080981595 0.171428571

Cost ($/kBtu yr)

51 1.3582

0.12

Heat exchanger cost ¼ 312.4A0.6 $/yr (A in ft2), boiler cost ¼ 135.9468Q0.7 $/yr (Q in kBtu/h) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Ciric and Floudas (1991) Papoulias and Grossmann (1983a) Floudas et al. (1986)a Dolan et al. (1989)

639,209 – 647,050 644,480

639,188 643,638 644,817

a

Using SI units. The original values are kH1C1 ¼ 0.1, kH2C1 ¼ 0.07, kH3C1 ¼ 0.06, kH4C1 ¼ 0.07, kH1C1 ¼ 0.055, and kH1CU ¼ 0.08, kH5CU ¼ 0.04 (kBtu/ft2°F). b

H5

3600 100

400 (12)

3642. 124

675

1705. 908

150

(15)

H2 H3 H4 H6

2527 450

1540

590 (11)

1912.5

540

115

(4.5)

5100

430

345

(60)

8750

300

230

(125)

7899.5 60

710

(47)

C1

(42.79313) (26.21572)

(10.82974)

Fig. 6.25 Optimal solution for Example H6C1, TAC ¼ 639,188$/yr (Ciric and Floudas, 1991).

292

Design and operation of heat exchangers and their networks

Example H4C4 This example was taken from He and Cui (2013), as is shown in Table 6.26, which was originally used by Grossmann and Sargent (1978). Luo et al. (2004) used this example for the optimal design of multistream heat exchanger networks. In their calculations, they took the overall heat transfer coefficient of 0.15 kBtu/ft2°F for all units. We find that their final network will also become the best design of the two-stream heat exchanger network even we use the problem data given in Table 6.26, which yields TAC ¼ 29,216$/yr after the revision. The corresponding network is shown in Fig. 6.26, in which there are two independent variables for stream splitting.

Table 6.26 Problem data for H4C4 (He and Cui, 2013). Stream

Tin (°F)

Tout (°F)

C_ (kBtu/h°F)

α (kBtu/ft2°F)

Cost ($/kBtu yr)

H1 H2 H3 H4 C1 C2 C3 C4 HU CU

470 450 370 310 200 150 185 140 456 100

320 240 150 200 420 400 330 300 456 180

22.4 17.5 28.5 20.1 16.8 23.2 35.1 17.25

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.6 0.3

11.05 5.31

Heat exchanger cost ¼ 35A0.6$/yr (A in ft2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Luo et al. (2004) for multistream HE Zhang et al. (2017)a He and Cui (2013)a Souto et al. (1992, Fig. 8)a,b

– 30,783 30,793 –c

29,216 30,777 30,794 30,869

a

No stream split. Using SI units. c After revision, H3CU and HUC3 are deleted. b

Optimal design of heat exchanger networks

293

Example H4C4—cont’d H1 H2 H3 H4

481.5

170

320

(43.80317)

(22.4)

3675

450

240

(17.5)

370

150

(16.24826)

(28.5)

310

200

(20.1)

21 200

420

(16.8)

400

150 (23.2)

3510

1808.5 330

185 (35.1)

2878.5 300

140 2760

2211

(17.25)

C1 C2 C3 C4

Fig. 6.26 Optimal solution for Example H4C4, TAC ¼ 29,216$/yr (Luo et al., 2004).

Example H4C5 This example is well known as the aromatic plant problem, which is a popular medium-scale problem and was firstly studied by Linnhoff and Ahmad (1990). The problem data are given in Table 6.27. The best Table 6.27 Problem data for H4C5 (He and Cui, 2013). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 C1 C2 C3 C4 C5 HU CU

327 220 220 160 100 35 85 60 140 330 15

40 160 60 45 300 164 138 170 300 250 30

100 160 60 400 100 70 350 60 200

0.5 0.4 0.14 0.3 0.35 0.7 0.5 0.14 0.6 0.5 0.5

60 6

Heat exchanger cost ¼ 2000 + 70A$/yr (A in m2) Continued

294

Design and operation of heat exchangers and their networks

Example H4C5—cont’d Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Pettersson (2005) Pava˜o et al. (2017a) Toffolo (2009) Pava˜o et al. (2017b) Pava˜o et al. (2017b)a Huo et al. (2013) Luo et al. (2009) Huo et al. (2012) Xiao et al. (2018)a Pava˜o et al. (2017c)a Lewin (1998) Bergamini et al. (2007)a Nu´n˜ez-Serna and Zamora (2016)a Peng and Cui (2015)a Huo et al. (2013)a Yan et al. (2009)a Yerramsetty and Murty (2008)a Lewin (1998)a Bogataj and Kravanja (2012)a Linnhoff and Ahmad (1990) Myankooh and Shafiei (2016)a Azeez et al. (2013) Zhu et al. (1995)a

2,904,953 2,919,154 2,919,684 2,919,675 2,920,763 2,922,600 2,922,298 2,925,634 2,927,431 2,928,629 2,938,000b 2,935,020 2,932,817 2,935,000 2,936,000 2,943,000 2,941,920 2,946,000 2,771,000 2,890,000 2,889,617 2,976,000 2,984,417

2,892,924 2,919,112 2,919,675 2,919,676 2,920,747 2,920,516 2,922,298 2,923,265 2,926,657 2,927,797 2,930,170 2,932,911

a

2,933,023 2,935,385 2,937,550 2,941,295 2,944,290 2,944,689 2,949,780 2,956,648 2,963,013 2,983,824

No stream split. After revision, HUC1 and the split of H1 are deleted.

b

network was obtained by Pettersson (2005), using a sequential match reduction approach based on an assignment model for the synthesis of large-scale heat exchanger networks. The network has seven stream splits and contains 15 independent variables. As is shown in Fig. 6.27, the network is similar to a hyperstructure, and there is a sequential use of hot utility for the cold stream C5. The revised TAC is 2,892,924$/yr.

Optimal design of heat exchanger networks

295

Example H4C5—cont’d (100)

H2

(160)

H3 H4

6013

1899 (24.41441)

14683.51

327

H1

40

4878 (83.46773)

220

160 3517

714.4022 (7.610622)

220

60

(60)

260

1231

(72.99186)

45

(400)

11715 300

22343 100 (100) (90.54818)

164

4721.730

35

2849.029

6104

2925.576

138

(70)

85 (350) (320.7601)

170 10744.14

1694 300

60 (60)

16651.40 5369.084

140 (200)

C1 C2 C3 C4 C5

(96.90193)

Fig. 6.27 Optimal solution for Example H4C5, TAC ¼ 2,892,924$/yr (Pettersson, 2005).

Example H5C5 This example is the well-known 10SP1 synthesis problem proposed by Pho and Lapidus (1973). Since then, a lot of researchers published their best solutions of this problem, which gradually approached to its global optimal design. The original problem data were given in English units (Pho and Lapidus, 1973, Table 2; Yee et al., 1990, Table 3). Since Papoulias and Grossmann (1983a), most researchers have used the problem data transferred from English units to international system of units (SI). For the unique comparison of the published network Table 6.28 Problem data for H5C5 (Lewin, 1998). Stream

Tin (K)

Tout (K)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 H5 C1 C2 C3 C4 C5 HU CU

433 522 544 500 472 355 366 311 333 389 509 311

366 411 422 339 339 450 478 494 433 495 509 355

8.79 10.55 12.56 14.77 17.73 17.28 13.9 8.44 7.62 6.08

1.704 1.704 1.704 1.704 1.704 1.704 1.704 1.704 1.704 1.704 3.408 1.704

37.64 18.12

Heat exchanger cost ¼ 145.63A0.6$/yr (A in m2) Continued

296

Design and operation of heat exchangers and their networks

Example H5C5—cont’d Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Pettersson (2008) Toffolo (2009) Lin and Miller (2004) Wei (2003), Wei et al. (2004a)a Gupta and Ghosh (2010) Huang and Karimi (2013) Floudas and Ciric (1989)a Linnhoff et al. (1979)a,c Athier et al. (1997)a,c He and Cui (2013)c Yee et al. (1990)a Peng and Cui (2015)c Huo et al. (2012) Pariyani et al. (2006)c Linnhoff and Flower (1978b)c Nishida et al. (1977)a,c Bergamini et al. (2007)c Escobar and Trierweiler (2013, Fig. A5) Nu´n˜ez-Serna and Zamora (2016, Fig. 8) Myankooh and Shafiei (2015) Yu et al. (2000)a,c Chen and Cui (2016)c Pho and Lapidus (1973)a,c Nu´n˜ez-Serna and Zamora (2016, Fig. 7)c Lewin et al. (1998)c Chakraborty and Ghosh (1999)c Papoulias and Grossmann (1983a, Fig. 9)c Su and Motard (1984)c

43,331 43,314 43,329 43,777 43,342 43,359 43,830b 43,857 43,856 43,392 43,878 43,411 43,431 43,439 43,934 43,984 43,220 43,570 43,646 43,718 43,176 43,656 44,158 43,733 43,452 44,214 – 44,124

43,314 43,321 43,329 43,342 43,359 43,376 43,393 43,407 43,411 43,429 43,440 43,469 43,523 43,590 43,627 43,646 43,649 43,688 43,692 43,698 43,733 43,751 43,803 44,074

a

Using English units. This value is obtained according to Fig. 9 of Floudas and Ciric (1989). The reported value of TAC was obviously a mistake. c No stream split. b

configurations, we have revised all the available results by the use of the problem data in SI, as is listed in Table 6.28. It shows that the effort for finding its global optimal design is just like an Olympic game. The best network configuration so far was obtained by Pettersson (2008), which has three independent variables for the three stream splits, as is shown in Fig. 6.28. According to his reported TAC of 43,331$/yr, the isothermal mixing of the splitting streams might be used. The local minimal TAC of 43,314$/yr was obtained by Toffolo (2009).

297

Optimal design of heat exchanger networks

Example H5C5—cont’d H1 H2 H3 H4 H5

588.93

433

366

(8.79)

522

341.3

762

(8.690022)

411

(10.55)

544

(4.985504)

422

(12.56)

1556.8

500

821.17 339

(14.77)

1300.3

472

339

(17.73)

1057.79

450

355

C1

(17.28)

366

478

C2

(13.9)

311

494 955.59

C3

(8.44)

333

433

C4

(7.62)

389

495

C5

(6.08)

67.75

(5.305256)

576.73

Fig. 6.28 Optimal solution for Example H5C5, TAC ¼ 43,314$/yr (Toffolo, 2009).

Example H6C4 This example was originally proposed by Ahmad (1985, p.146, Table 5.3). It has been used as a benchmark by many other researchers. In most publications, the data listed in Table 6.29 have been used, in which the thermal capacity rate of stream C3 and the target temperature of stream H5 have been changed from the original values of 195 kW/K and 85°C to 180 kW/K and 86°C, and the outlet temperatures of hot and cold utilities that were not mentioned by Ahmad have been supplemented with 198°C and 20°C, respectively. In this example, all units have the same overall heat transfer coefficient, and the heat exchanger cost is proportional to the area for all units. That means, the optimal network configuration can be obtained according to the composite curves at the optimal minimum temperature difference, which consists of a lot of heat exchangers and stream splits. Therefore, this example is not suitable for testing new synthesis procedures being developed, and the restriction of Continued

298

Design and operation of heat exchangers and their networks

Example H6C4—cont’d Table 6.29 Problem data for H6C4 (Khorasany and Fesanghary, 2009). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 H5 H6 C1 C2 C3 C4 HU CU

85 120 125 56 90 225 40 55 65 10 200 15

45 40 35 46 86 75 55 65 165 170 198 20

156.3 50 23.9 1250 1500 50 466.7 600 180 81.3

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

100 15

Heat exchanger cost ¼ 60A$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Own work Xiao and Cui (2017) Peng and Cui (2015) Chen and Cui (2016) Zhang et al. (2017) He and Cui (2013) Ravagnani et al. (2005) Myankooh and Shafiei (2016) Myankooh and Shafiei (2015) Huang et al. (2012)c Huo et al. (2013) Khorasany and Fesanghary (2009) Gorji-Bandpy et al. (2011) Huo et al. (2012) Yerramsetty and Murty (2008)

– 5,589,493a 5,596,079 5,593,970b 5,607,761 5,609,271 5,672,821 5,640,721 5,640,724 5,738,268 5,645,688 5,662,366 5,139,495 5,657,000 5,666,756

5,585,391 5,585,784 5,592,285 5,593,028 5,607,754 5,609,270 5,640,719 5,642,268 5,642,387 5,642,958 5,643,660 5,646,818 5,666,750 Total annual cost ($/yr)

Solutions in the literature Heat exchanger cost 5 8000 + 60A$/yr (A in m2)

Reported

Revised

Pava˜o et al. (2017a) Pava˜o et al. (2017b) Huang and Karimi (2014)

5,713,267 5,715,026 5,733,679

5,713,267 5,714,929 5,734,428

a

After revision, H1C2 is deleted. After revision, 5 units are deleted. With stream split.

b c

299

Optimal design of heat exchanger networks

Example H6C4—cont’d 3457.823

85

H1

1551.551 258.719

(156.3)

120

H2

770

125 (23.9)

H4

(1250)

45

1190.599

999

(50)

H3

1717

1077.618

382.132 50.490

384.361

40

564 35

56

46

3679

90

H5

(1500)

225

H6

574.632 701.430

12500

2321

4342.490

1301

580.660

86 75

(50)

40

55

(466.7)

55

65

(600)

12956 165

65 (180)

7430

10

170

(81.3)

C1 C2 C3 C4

(A) 2315

H1 H2

85

40

(50)

2151

125 (23.9)

H4

(1250)

H6

2857

1143

120

H3

H5

45

(156.3)

35

12500

56

46

2936

90

3064

86

(1500)

5579.207

225

1921

75

(50)

55

40

3936.783 65

(466.7)

55 (600)

165

12421

(204.1828)

65 (180)

170

10 (81.3)

(B)

8000

C1 C2 C3 C4

(30.81177)

Fig. 6.29 Optimal solutions for Example H6C4. (A) Heat exchanger cost ¼ 60A$/yr (A in m2), TAC ¼ 5,585,391$/yr. (B) Heat exchanger cost ¼ 8000 + 60A$/yr (A in m2), TAC ¼ 5,713,267$/yr (Pavão et al., 2017a).

no stream splitting was applied to this example by many researchers. The best network without stream splitting is shown in Fig. 6.29A, which consists of 22 units and contains 12 independent variables, with minimum TAC of 5,585,391$/yr. To make the example more meaningful, Huang and Karimi (2014) modified the heat exchanger costs by adding the fixed cost of 8000$ instead of zero, and the stream splitting was allowed. The best solution of the modified example was obtained by Pava˜o et al. (2017a), which has 12 units, two stream splits, and four independent variables, as is shown in Fig. 6.29B.

300

Design and operation of heat exchangers and their networks

Example H6C5 This example is taken from Silva et al. (2010), which was a real industrial case first investigated by Castillo et al. (1998) by means of pinch analysis. The problem data are provided in Table 6.30. The best network configuration shown in Fig. 6.30 was reported by Stegner et al. (2014), which contains one stream split and one independent variable, and the achieved TAC is 139,407$/yr.

Table 6.30 Problem data for H6C5 (Silva et al., 2010). Stream

Tin (K)

Tout (K)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 H5 H6 C1 C2 C3 C4 C5 HU CU

1113 349 323 453 453 363 297 298 308 363 453 503 293

313 318 313 350 452 318 298 343 395 453 454 503 313

4.9894 4.684 0.772 0.6097 292.7 3.066 329.8 0.5383 3.727 0.6097 2581.1

1.5 1.5 1.5 1.5 0.8 1.5 0.8 1.5 1.5 1.5 0.8 1.5 0.8

110 15

Heat exchanger cost ¼ 9094 + 485A0.81$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Stegner et al. (2014) Pava˜o et al. (2017b) Wang et al. (2017)a Silva et al. (2010) Pava˜o et al. (2016)

139,400 139,453 139,775 139,777 140,068

139,407 139,491 139,775 140,141

a

No stream split.

301

Optimal design of heat exchanger networks

Example H6C5—cont’d H1 H2 H3 H4

1113

1323.7

2581.1 54.9 31.9

313

(4.9894)

145.2

349

318

(4.684)

7.7

323

313

(0.772)

23.9 38.9

453

350

(0.6097)

453

H5

(292.7)

H6

(3.066)

363

292.4 0.3

318

138.0

452 297

298

C1

(329.8)

343

(147.9407)

298 (0.5383)

395

308

C3

(3.727)

453

363 (0.6097)

454

C2

453 (2581.1)

C4 C5

Fig. 6.30 Optimal solution for Example H6C5, TAC ¼ 139,407$/yr (Stegner et al., 2014).

Example H8C7 This example is given in Table 6.31 and was investigated by Bjork and Pettersson (2003). They reported that their heat exchanger network configuration had an objective of 1,513,854$/yr and was divided into three subproblems of five streams in each subsystem, but the network structure was not given in their publication. The best network configuration shown in Fig. 6.31 was obtained by Pava˜o et al. (2017a), which has two stream splits and contains five independent variables. The TAC reaches 1,497,252$/yr.

Continued

302

Design and operation of heat exchangers and their networks

Example H8C7—cont’d Table 6.31 Problem data for H8C7 (Bjork and Pettersson, 2003). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 H5 H6 H7 H8 C1 C2 C3 C4 C5 C6 C7 HU CU

180 280 180 140 220 180 200 120 40 100 40 50 50 90 160 325 25

75 120 75 40 120 55 60 40 230 220 190 190 250 190 250 325 40

30 60 30 30 50 35 30 100 20 60 35 30 60 50 60

2 1 2 1 1 2 0.4 0.5 1 1 2 2 2 1 3 1 2

80 10

Heat exchanger cost ¼ 8000 + 500A0.75$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Pava˜o et al. (2017a) Escobar and Trierweiler (2013) Pava˜o et al. (2017b) Fieg et al. (2009) Wang et al. (2017)a Pava˜o et al. (2017c)a Peng and Cui (2015)a Myankooh and Shafiei (2015)a

1,497,325 1,506,667 1,507,290 1,510,891 1,519,250 1,525,394 1,527,240 1,460,097

1,497,252 1,503,317 1,506,858 1,510,892 1,519,250 1,523,736 1,526,948 1,587,206

a

No stream split.

Optimal design of heat exchanger networks

303

Example H8C7—cont’d H1 H2 H3

75

(30)

2400

280

7200

120

(60)

3150

180

75

(30)

H4

140

H5

220

H6

180

H7

200

H8

3150

180

1179

1821.014

40

(30)

853 (10.29220)

120

(50)

4375

55

(35)

4200

60

(30)

852.769

120 (100)

864.645

6283 40

1126 40

230

(20)

100

220

(60)

190

875

40 (35)

190 250

50 (30)

4835

50 (60) (30)

190 250

3000

90 4147

(50)

160 (60)

C1 C2 C3 C4 C5 C6 C7

Fig. 6.31 Optimal solution for Example H8C7, TAC ¼ 1,497,252$/yr (Pavão et al., 2017a).

Example H6C10 This example is a real industrial-sized problem used by Khorasany and Fesanghary (2009). In their problem data, the inlet and outlet temperatures of the cold utility were given as 311°C and 355°C (also in Gorji-Bandpy et al. (2011)), which was obviously a typing error. Brandt et al. (2011) believed that the values should be 31.1°C and 35.5°C according to the reported network structure and TAC of Khorasany and Fesanghary (2009). However, Huo et al. (2013) thought that they should be 311 K and 355 K (38°C and 82°C), and their problem data were used by other researchers, as shown in Table 6.32. Using the monogenetic algorithm (Fieg et al., 2009), we have obtained the best network shown in Fig. 6.32, which has eight independent variables and the minimum TAC of 6,673,406$/yr. Continued

304

Design and operation of heat exchangers and their networks

Example H6C10—cont’d Table 6.32 Problem data for H6C10 (Huo et al., 2013). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 H5 H6 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 HU1 HU2 CU

385 516 132 91 217 649 30 99 437 78 217 256 49 59 163 219 1800 509 38

159 43 82 60 43 43 385 471 521 418.6 234 266 149 163.4 649 221.3 800 509 82

131.51 1198.96 378.52 589.545 186.216 116 119.1 191.05 377.91 160.43 1297.7 2753 197.39 123.156 95.98 1997.5

1.238 0.546 0.771 0.859 1 1 1.85 1.129 0.815 1 0.443 2.085 1 1.063 1.81 1.377 1.2 1 1

35 27 2.1

Heat exchanger cost ¼ 26,600 + 4147.5A0.6$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Own work Brandt et al. (2011) Zhang et al. (2017)b Pava˜o et al. (2017c)b Huo et al. (2013, Fig. 8)b Khorasany and Fesanghary (2009) Gorji-Bandpy et al. (2011)b

– 6,110,902a 6,511,584 7,301,437 7,385,856 7,435,740 8,220,154

6,673,406 6,790,990 7,094,611 7,128,572 7,218,412 7,982,270 9,048,250

a

Inlet/outlet temperatures of cold utility: 31.1°C/35.5°C. No stream split.

b

305

Optimal design of heat exchanger networks

Example H6C10—cont’d H1

12857 (78.5403)

385

12270

159 (186.0371) (402.1666) (273.3042) (149.7297)

(131.51)

516

H2

(1198.96)

H3

(378.52)

H4

(589.549)

H5

(186.216)

H6

(116)

324812 43 18926 82

132

18276 60

91

32402 43

217 43986.12

649

18840 43

7469

30

385

(119.1)

42281

99

471

(191.05)

71071

7033 521 HU1

437 (377.91)

24711.93

78

418.6

(160.43)

54642 234

217 (1297.7)

22061 266

256 27530

149

(2753)

49 (197.39)

59

163.4

(123.156)

2660 649 HU1

163 (95.98)

221.3

219 4594

(1997.5)

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10

Fig. 6.32 Optimal solution for Example H6C10, TAC ¼ 6,673,406$/yr.

Example H10C10 As a large-scale heat exchanger network, this example was used by Luo et al. (2009), which was taken from Xiao et al. (2006) for the synthesis of heat exchanger network using multistream heat exchangers. The problem data are listed in Table 6.33. The best network shown in Fig. 6.33 was obtained by Bohnenstaedt et al. (2014), which has one stream split and five independent variables and reaches the minimum TAC of 1,716,695$/yr. Continued

306

Design and operation of heat exchangers and their networks

Example H10C10—cont’d Table 6.33 Problem data for H10C10 (Luo et al., 2009). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 HU CU

180 280 180 140 220 180 170 180 280 180 40 120 40 50 50 40 40 120 40 60 325 25

75 120 75 45 120 55 45 50 90 60 230 260 190 190 250 150 150 210 130 120 325 40

30 15 30 30 25 10 30 30 15 30 20 35 35 30 20 10 20 35 35 30

2 0.6 0.3 2 0.08 0.02 2 1.5 1 2 1.5 2 1.5 2 2 0.06 0.4 1.5 1 0.7 1 2

70 10

Heat exchanger cost ¼ 8000 + 800A0.8$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Bohnenstaedt et al. (2014) Pava˜o et al. (2017b) Zhang et al. (2016a,b)a Laukkanen et al. (2012) Luo et al. (2009) Pava˜o et al. (2017c)a Myankooh and Shafiei (2015)a Laukkanen and Fogelholm (2011)

1,717,295 1,725,295 1,731,679 – 1,753,271 1,763,488 1,762,280 1,811,900

1,716,695 1,725,283 1,731,679 1,738,335 1,753,271 1,758,141 1,762,280 1,793,534

a

No stream split.

307

Optimal design of heat exchanger networks

Example H10C10—cont’d H5 H9

1100(10.61347)

220

120 2050

280

190

50 (30)

40 40

150 210

(20)

1400

1100

120 (35)

2400

280

C6

H3

180

H4

75 40

1800

140

1050

744

1456.201

60

H7

(30)

40 (20)

3444

170

60

120 (35)

(10)

1753

C2

55

1503

H8

C10

45 50 (20)

1400

3900

180

C5

50

(30)

1250 180

2246.502

(30)

250

C1

C9

45

(30)

180

260

(35)

(30)

120

120

C4

(30)

130

C7 C8

3150

(15)

230

H6

75

(30)

1050

800 90

(10)

H10

3150

180

(15)

150

H2

H1

(25)

190

1350

40 (35)

C3

Fig. 6.33 Optimal solution for Example H10C10 (Bohnenstaedt et al., 2014), TAC ¼ 1,716,695$/yr.

Example H13C7 This large-scale heat exchanger network example was taken from Escobar and Trierweiler (2013) who used the problem data of Sorsak and Kravanja (2002) for synthesis of heat exchanger networks comprising different heat exchanger types, however, with their own equipment cost equation, as is given in Table 6.34. In this example, the inlet temperature of one of the hot stream (H13, Tout ¼ 1034.5°C) is higher than the hot utility temperature (927°C). Consequently, H13 shall be matched with the cold stream C7, which has the highest target temperature (923.78°C). Using the monogenetic algorithm (Fieg et al., 2009) and taking the network configuration of Xiao et al. (2018) for the initial subnetworks, we had obtained the heat exchanger network better than previous optimal results. However, we noticed that in the obtained network, H13 is heated again to reach its target temperature. This is because of the fact that the monogenetic algorithm is based on the stagewise superstructure of Yee et al. (1990), in which the utilities are located at the outlets of the network if necessary. To avoid the reheating of hot streams, we manually deleted the match H13HU and added the heater before the match of H13C7. The best network configuration is shown in Fig. 6.34, which has three independent variables. After local optimization of the three variables, the TAC reaches to 1,410,649$/yr. Continued

308

Design and operation of heat exchangers and their networks

Example H13C7—cont’d Table 6.34 Problem data for H13C7 (Xiao et al., 2018). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 C1 C2 C3 C4 C5 C6 C7 HU CU

576 599 530 449 368 121 202 185 140 69 120 67 1034.5 123 20 156 20 182 318 322 927 9

437 399 382 237 177 114 185 113 120 66 68 35 576 343 156 157 182 318 320 923.78 927 17

23.1 15.22 15.15 14.76 10.7 149.6 258.2 8.38 59.89 165.79 8.74 7.62 21.3 10.61 6.65 3291 26.63 31.19 4011.83 17.6

0.06 0.06 0.06 0.06 0.06 1 1 1 1 1 1 1 0.06 0.06 1.2 2 1.2 1.2 2 0.06 5 1

Cost ($/kW yr)

Heat exchanger cost ¼ 4000 + 500A0.83$/yr (A in m2) Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Own work Xiao et al. (2018)a Escobar and Trierweiler (2013) Pava˜o et al. (2016)

– 1,447,482 1,461,006b 1,516,482

1,410,649 1,435,931 1,489,667

a

No stream split. After revision, H8C1 is deleted.

b

1047.2

121 1251.84

202

H7

(258.2)

H8

(8.38)

H9 H10

H12

153.44

185

3137.56

185

449.92

113 1197.8

140

120 497.37

69

(23.1)

H2

(15.22)

H3

(15.15)

H13

(59.89)

576

H1

66

3210.9

437 3044

599

399 473.4 1768.8 382

530 1034.5

9766.1

576

(21.3)

318

320

(165.79)

(4011.83)

454.48

120

351.8

923.78

68

322

(8.74)

(17.6)

243.84 35

67

H4

(7.62)

156

20 (6.65)

157

156 (3291)

182

76.01

20 (26.63)

C2 C3

H5

3129.1

449

237

544.2624

368

(3.295989)

177

(10.7)

343

834.8

123

318

(10.61)

182 (31.19)

568.5

Fig. 6.34 Optimal solution for Example H13C7, TAC ¼ 1,410,649$/yr.

C7

(14.76)

1499.4

C4

C6

(7.091527)

C1 C5

Optimal design of heat exchanger networks

H11

114

(149.6)

Example H13C7—cont’d

H6

309

Example H22C17 This example was first used by Bjork and Pettersson (2003). It is the largest heat exchanger network available in the literature and being investigated by many researchers. The problem data are listed in Table 6.35. We used the Table 6.35 Problem data for H22C17 (Bjork and Pettersson, 2003). Stream

Tin (°C)

Tout (°C)

C_ (kW/K)

α (kW/m2 K)

Cost ($/kW yr)

H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 H16 H17 H18 H19 H20 H21 H22 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 HU CU

180 280 180 140 220 180 170 180 280 180 120 220 180 140 140 220 220 150 140 220 180 150 40 120 40 50 50 40 40 120 40 60 50 40 120 40 50 50 30 325 25

75 120 75 45 120 55 45 50 90 60 45 120 55 45 60 50 60 70 80 50 60 45 230 260 190 190 250 150 150 210 130 120 150 130 160 90 90 150 150 325 40

30 15 30 30 25 10 30 30 15 30 30 25 10 20 70 15 10 20 70 35 10 20 20 35 35 30 60 20 20 35 35 30 10 20 35 35 30 30 50

2 2.5 2 2 1.5 2 2 2 2 2 2 2 2 2 2 2.5 2.5 2 2 2 2 2.5 1.5 1 1.5 2 2 2 2 2.5 2.5 2.5 3 1 1 1.75 1.5 2 2 1 2

70 10

Heat exchanger cost ¼ 8000 + 800A0.8$/yr (A in m2)

Example H22C17—cont’d Total annual cost ($/yr) Solutions in the literature

Reported

Revised

Own work Pava˜o et al. (2017b) Xiao et al. (2018)a Huang and Karimi (2014) Zhang et al. (2016b)a Ernst et al. (2010) Zhang et al. (2016a)a Pettersson (2005) Escobar and Trierweiler (2013) Bjork and Pettersson (2003)

– 1,900,614 1,936,288 1,937,377 1,939,149 1,943,536 1,954,417 1,998,000 2,055,421 2,073,251

1,897,159 1,900,577 1,936,287 1,938,180 1,939,107 1,943,536 1,953,721 1,967,465 2,050,327 –b

a

No stream split. Network structure was not given.

b

H2 H10

2400

280

120

H4

(15)

1400 2200

180

H12 H16

220

120

(25)

1451.545

854.668

H5

244

50 (60)

(26.83629) (15.43829)

160

(17.74452)

120 (35)

1400 1750

180

75

3500

120 (35)

90

40 (35)

1250

180

950

120

40 (20)

H9

280

H18

150

2200

1550

H15

40

180

3000

900

1200

700

(30)

140

50

140

50 3150

2450

220

(70)

40

220

5250

50

(35)

70

190

40

180

H19

140

(20)

C1

H21

40 1000

150

80

150

H22

(70)

150

200

30 (50)

(40.48405)

180 (30)

1050

75 50

190 3150

(30)

60 50

1800

300

C11

(20)

40 2250

H11

(10)

45

130

C17

C3

200

(10)

60

(10)

4200

C9

700

(35)

2200 1600 1600

C15

60

(20)

230

H17

(30)

(35)

H20

C16

45

(20)

650

(15)

C7

50

130

C6

C8

(30)

90

90

(35)

(30)

C14 H14

C10

45

150

C2

(10)

(8.234862)

170

300

150

H1

856

(20)

H8

55

(30)

120

150

55

(10)

180

H7

C13

(30)

260

H13

C5

206

(25)

(15)

2594

H6

60 2294.379

220 210

50

250

45

120

60

2500

1050

(30)

(30)

220

H3

1800

140

120

(20)

C12

45

(30)

C4

Fig. 6.35 Optimal solution for Example H22C17, TAC ¼ 1,897,159$/yr. Continued

312

Design and operation of heat exchangers and their networks

Example H22C17—cont’d monogenetic algorithm (Fieg et al., 2009) to solve this problem and obtained the best heat exchanger network shown in Fig. 6.35, which consists of 16 subnetworks. The largest subnetwork comprises four hot streams and two cold streams. There are totally eight independent variables, which yields the minimum TAC of 1,897,159$/yr.

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