Optimal control process of heat exchanger networks

CHAPTER 9

Optimal control process 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 b Institute of Thermodynamics, Gottfried Wilhelm Leibniz University Hannover, Hannover, Germany c Institute of Thermal Energy and Environmental Engineering, Tongji University, Shanghai, China

The design of heat exchanger networks (HEN) has been discussed in Chapter 6. Here, we will discuss the topic of the automatic control problem for the operation of HEN. Compared with synthesis of nominal and flexible HENs, less effort has been dedicated to find methods for the operation of a HEN and its control. The concern on operation and control arises when they have to deal with significant changes in the operating conditions such as disturbances of inlet parameters and operation state changeover. A HEN is flexible when it is capable of absorbing long-term variations on inlet stream conditions or having the capability of changing stream temperature targets significantly. This concept has been discussed by Floudas and Grossmann (1986), Calandranis and Stephanopoulos (1986), Kotjabasakis and Linnhoff (1986), and Galli and Cerda (1991). On the other hand, controllability is associated with short-term perturbations, stability, and safe transitions from one operating point to another. Morari (1983) used a term “resilience” to describe the ability of a system to move fast or smoothly from one operating condition to another, including start-up and shutdown, and to deal effectively with disturbances. He also divided the problems of process systems into two topics: static resilience and dynamic resilience. The static resilience refers to the ability of a system to handle different feedstocks, product specifications, operating conditions, etc. in the steady state, while the dynamic resilience is concerned with the transient behavior in the event of a changeover or when the disturbances enter the plant. Therefore, according to Morari’s definition, resilience involves all the meaning of flexibility, operability, and controllability. Therefore, a resilient synthesis means that the flexibility, operability, and controllability of a HEN will be considered simultaneously. Design and Operation of Heat Exchangers and their Networks https://doi.org/10.1016/B978-0-12-817894-2.00009-1

© 2020 Elsevier Inc. All rights reserved.

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Design and operation of heat exchangers and their networks

9.1 Feasibility, flexibility, and controllability of heat exchanger networks 9.1.1 Feasibility and flexibility A well-designed HEN or multistream heat exchanger should be a flexible one that is able to adapt to parameter deviations from nominal values and remain flexible when the parameters vary within a prespecified range. There is another word, feasibility, which is easily confounded with flexibility. Feasibility analysis gives us the information on whether or not a HEN can maintain the target temperatures after any possible disturbances within a prespecified range. However, flexibility analysis will tell us how the operating cost changes over the specified range of uncertain parameters. Swaney and Grossmann (1985a,b) proposed a quantitative index, which they called “flexibility index” to measure the size of the parameter space over which feasible steady-state operation of the plant could be obtained by proper regulation of controlling variables. Later, Roetzel and Luo (2002) defined a new flexibility factor to quantitatively address the flexibility of a HEN. Assume a HEN in a process, its thermal performance at the nominal operation point is described by hðd, cN , uN Þ ¼ 0

(9.1)

gðd, cN , uN Þ  0

(9.2)

where d is the vector of design variables and cN and uN are the vectors of controlling variables and uncontrolled operation variables at the nominal operation point, respectively. The set h contains the exit stream temperatures to be maintained at given targets, and the set g contains the exit stream temperatures to be limited in given regions. The total utility cost at the nominal operation point is calculated according to X X CN ¼ fHU ðQHU, N Þ + fCU ðQCU, N Þ (9.3) in which fHU and fCU are cost functions of corresponding hot and cold utility requirements, respectively. A HEN is feasible if for any deviations in uncontrolled operation variables, Δu ¼ u  uN (u 2 Ru), there exists c 2 Rc such that the constraints h(d, cN, uN) ¼ 0 and g(d, cN, uN)  0 are satisfied. To specify the feasibility of a HEN quantitatively, the flexibility index D introduced by Swaney and Grossmann (1985a,b) is adopted, which is defined as DðdÞ ¼ maxδ

(9.4)

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Optimal control process of heat exchanger networks

  s:t: 8u 2 Ru  uN  δΔu  u  uN + δΔu + 9c ½hðd, c, uÞ ¼ 0, gðd, c, uÞ  0

where Δu and Δu+ are the vectors of maximum possible deviations of the uncontrolled operation variables in positive and negative directions, respectively. Then, the feasibility can be stated as follows: A HEN is feasible, if the flexibility index D 1. It indicates how large the deviation range of the uncontrolled operation variables can be extended. This index might be named as feasibility index rather than flexibility index because it does not give any information about the variation of energy recovery (utility cost). If the HEN is feasible, for a given u 2 Ru, an optimal controlling vector c 2 Rc can be found such that the total utility cost reaches the minimum: n X o X (9.5) min C ðd, c, uÞ ¼ min a QHU ðd, c, uÞ + b QCU ðd, c, uÞ c2 Rc

c2 Rc

s:t: hðx, c, uÞ ¼ 0 gðx, c, uÞ  0 If there is no solution of c 2 Rc, which satisfies the constraints, the HEN is not feasible at the point u, and we set Cmin(d, u) ! ∞. A HEN is flexible if it is feasible, and for any possible disturbances in u 2 Ru, the running cost does not increase dramatically. To evaluate the flexibility of a HEN quantitatively, Roetzel and Luo (2002) introduced a flexibility factor defined as the ratio of the minimum total utility cost at the nominal operation point Cmin(d, uN) to the maximum value of the minimum total utility cost in the range of all possible values of the uncontrolled operation variables: F ðdÞ ¼

Cmin ðd, uN Þ max Cmin ðd, uÞ

(9.6)

u2 Ru

This factor can well describe the flexibility of the HEN: F ¼ 1 means that the disturbances have no influence on the utility cost; F ¼ 0 indicates a nonfeasible HEN. If F falls below a given critical value Fcr, then the exchanger is considered to be nonflexible. Example 9.1 Flexibility analysis of a heat exchanger network This example is firstly given by Floudas and Grossmann (1987) and then reanalyzed by Roetzel and Luo (2002). The problem data, network construction, and heat transfer areas of the heat exchangers are given in Table 9.1, Fig. 9.1, and Table 9.2, respectively. The hot utility cost (300°C)

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Design and operation of heat exchangers and their networks

is 171.428  104 $/kWh, and the cold utility cost (30–50°C) is 60.576  104 $/kWh. All heat exchangers including the heater and cooler are of counterflow type. For the variation range of operation conditions given in Table 9.1, determine the flexibility index and flexibility factor. Solution 00 Let ti be the temperature of the ith fluid at the exit of the network before 000 entering a heater or cooler, and ti be that at the exit of the network after leaving a heater or cooler. If there is no heater or cooler at the network 000 00 exit of the ith fluid, we have ti = ti . For the case using a heater at the ith exit of the network, it requires Table 9.1 Nominal operation conditions and deviations of Example 9.1. Stream

C_ (kW/K)

t0 (°C)

t00 (°C)

H1 H2 C1 C2

1.4  0.4 2.0 3.0 2.0  0.4

310  10 450 40 115  5

50 280 120 280

H2

c1

2

CU

H2

Ch1 Ch2

c2 C2

2

HU

E3 c3

4

C2

4

Ch3 Ch4 H1

E2

1

1

Ch5 Ch6 C1

E1 c5

3

H1

c4

3

C1

Fig. 9.1 Network configuration for the example. Table 9.2 Overall heat transfer coefficients and areas of heat exchangers in Example 9.1. Heat exchanger

Match

k (kW/m2K)

A (m2)

E1 E2 E3 HU CU

H1-C1 H1-C2 H2-C2 HU-C2 H2-CU

0.8 0.8 0.8 0.8 0.8

103.83 56.583 23.315 23.770 13.013

Optimal control process of heat exchanger networks 00 ti00  tub,i 0 00 000 tlb, i  ti  0 00 0 ti000  tub,i

t00ub,i

(9.7) (9.8) (9.9)

t00lb,i are

where and upper and lower bound target temperatures of the ith fluid. Then, we can regulate the flow rate of the heating medium or the bypass of the working fluid to maintain the fluid temperature within the target temperature range. Similarly, for the case using a cooler at the ith exit of the network, the following constraints should be met: 00 00 tlb, i  ti  0 00 000 tlb, i  ti  0 00 0 ti000  tub,i

(9.10) (9.11) (9.12)

According to Fig. 9.1, the network has four streams (H1, H2, C1, and C2) and three heat exchangers (E1, E2, and E3) containing six channels (Ch1–Ch6). There are five bypasses that correspond to five controllers with the bypass thermal flow rates ci (i = 1, 2, …, 5) as the control variables. This network does not contain loops and can be calculated explicitly as follows: 0 0 0 ¼ tH1 , tc,0 E2 ¼ tC2 For E2 : C_ h,E2 ¼ C_ H1  c4 , C_ c,E2 ¼ C_ C2  c2 , th,E2

For E1 : C_ h,E1 ¼ C_ H1 , C_ c,E1 ¼ C_ C1  c5 , 00 c4 t 0 þ C_ h, E2 th, E2 0 0 0 ¼ H1 , tc,E1 ¼ tC1 th,E1 C_ h,E1 For E3 : C_ h, E3 ¼ C_ H2  c1 , C_ c,E3 ¼ c2 þ c3 , 0 0 0 th,E3 ¼ tH2 , tc,E3 ¼

0 00 c2 tC2 þ c3 tc,E2 C_ c, E3

For HU: C_ h,HU ! ∞, C_ c,HU ¼ C_ C2 , t 0 h, HU ¼ 300°C,   00 00 þ C_ C2  C_ c, E3 tc,E2 C_ c,E3 tc,E3 0 tc, HU ¼ C_ c,HU For CU: 0  00  tH2 C_ H2 th,CU , C_ h,HU ¼ C_ H2 , C_ c,CU ¼ 50  30 0 00 c1 tH2 þ C_ h,E3 th, E3 0 0 ,t c,CU ¼ 30°C th, CU ¼ C_ h,CU

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Design and operation of heat exchangers and their networks

The outlet temperatures of the jth exchanger ( j = E1, E2, E3, HU, and CU) are obtained from Eq. (6.2):  0 00 0 (9.13) th, j ¼ 1  εj th, j þ εj tc, j  0 00 0 (9.14) tc, j ¼ Rj εj th, j þ 1  Rj εj tc, j where NTUj ¼ kj AE, j =C_ h,E, j

(9.15)

Rj ¼ C_ h,E, j =C_ c,E, j

(9.16)

εj ¼

1  eNTUj ð1Rj Þ 1  R eNTUj ð1Rj Þ

(9.17)

j

Attention should be payed to the following special cases: For NTUj →∞, Rj  1 : εj ¼ 1

(9.18)

For NTUj →∞, Rj > 1 : εj ¼ 1=Rj NTUj For Rj ¼ 1 : εj ¼ 1 þ NTUj

(9.19)

For Rj ¼ 0 : εj ¼ 1  eNTUj

(9.21)

For Rj →∞ : εj ¼ 0, Rj εj ¼ 1  e

NTUc, j

(9.20)

(9.22)

where NTUc, j ¼ kj AE, j =C_ c,E, j

(9.23)

The optimal control vector c can be found by minimizing the utility cost C(d, c, u):  min 171:428  104 C_ C2 ðtc, out, HU  tc,in, HU Þ c2Rc

þ60:576  104 C_ H2 ðth,in,CU  th,out, CU Þg

(9.24)

00  th,out, E1 ¼ 0 tH1 00 tH2  th,in, CU  0

s:t:



00 tH2  th,out, CU ¼ 0  0 00 C_ c, E1 tc,out, E1 þ c5 tC1 ¼0 =C_ C1  tC1 00 tc,in, HU  tC2 0 00 tc,out, HU  tC2 ¼0 c2 þ c3  C_ C2  0

  Rc ¼ 0  c1  C_ H2 , 0  c2  C_ C2 , 0  c3  C_ C2 , 0  c4  C_ H1 , 0  c5  C_ C1

By solving Eq. (9.24) at the nominal operation point, the minimum total utility cost can be obtained as Cmin(d,uN) = 0.8117 $/h. The corresponding

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437

control variables are c1 = 0.4242 kW/K, c2 = 0.2685kW/K, c3 = 0.7356 kW/K, c4 = 0.7594 kW/K, and c5 = 0.9678 kW/K. The flexibility index can be obtained by solving Eq. (9.4), which yields D = 1. The worst operation condition lies at cmin(umax). By substituting the utility cost at this point and that at the nominal operation point into Eq. (9.6), the flexibility factor is obtained as F = 0.2775. That means, although the network is feasible, the utility cost at the worst operating point would be 2.925 $/h, which is 260% higher than that at the nominal operation point.

9.1.2 Operability and controllability analysis In this book, operability considerations mainly deal with manipulation methodology to maintain the target temperatures under uncertainties or operation changes between different steady states so that the utility consumption is minimized; therefore, it is closely linked to structural network flexibility. Controllability is to deal with the maintaining target output parameters upon short-term deviations of inlet parameters and stable and safe transitions from one operating point to another. 9.1.2.1 Operability analysis The operability analysis (Aguilera and Marchetti, 1998) is usually based on the existing structural information. It can be also incorporated into the synthesis/retrofit process. Consider a HEN with N streams exchanging heat in it, there are NU (NU  N) process streams need to be heated or cooled to their target output temperatures. The supply temperatures and the mass flow rates of process streams are considered as the inlet variables, and the properties of process streams are all taken as constants. Suppose that there are NM + NT (NM + NT < 2 N) regulatable parameters, in which NM represents mass flow rates and NT represents temperatures, respectively. The remainders are unregulatable parameters, NU ¼ 2 N  (NM + NT). When some of them deviate from their design values, the NM + NT inlet parameters (or some of them) are needed to be regulated to maintain the target output temperatures. There will be three cases affecting the operability of the system (Li et al., 2002): (1) NM + NT ¼ NU For this case, there will exist only one set of regulation solution to exactly maintain all the target output temperatures. Although some of the inlet temperatures could be regulatable parameters, in practical processes, the inlet flow rates are much easier to be regulated than the inlet temperatures of

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Design and operation of heat exchangers and their networks

the streams. But if all the target output temperatures are only required to maintain within the limited ranges, there may exist infinite numbers of regulation solutions. Thus, the optimal solution can be determined, and the task becomes an optimization problem. (2) NM + NT > NU For this case, if the deviations are within the feasible region of the HEN, there will exist infinite large number of regulation solutions, with which all the target output temperatures can be exactly maintained or can be maintained within limited ranges. Therefore, the optimal set need to be determined. (3) NM + NT < NU For this case, only some of the target output temperatures of NU process streams can be exactly maintained even if the deviations are within the feasible region, but there may still exist infinite number of regulation solutions to maintain all the target output temperatures of NU process streams within their limited ranges. Thus, the optimal regulation solution also needs to be determined. 9.1.2.2 Controllability analysis A HEN is controllable when there are enough regulatable bypasses connected with these NM + NT inlets; therefore, controllability should be considered in the synthesis stage instead of for an existing structure. How to manipulate the bypass values within limited time is a task of control design. Controllability includes two aspects: structural controllability and dynamic controllability. The former is often expressed as criteria or index describing relations between disturbances and controlled variables, and the latter is expressed as criteria based on dynamics model of HEN. From control theory, a system is controllable if there exists a control signal c(τ) such that the state of the system can be taken from any initial state x0 to any desired final state xj in a finite time interval. Papalexandri and Pistikopoulos (1994a) gave an explicit structural controllability criteria based on a multiperiod hyperstructure network representation; they included their criteria within a synthesis/retrofit mixed-integer nonlinear programming model with total annualized cost as objective function.

9.2 Synthesis and retrofit design of flexible and controllable heat exchanger networks Since the mid 1980s several authors have investigated flexibility of HENs, for example, Kotjabasakis and Linnhoff (1986) that introduced sensitivity tables to

Optimal control process of heat exchanger networks

439

find which heat exchanger areas should be increased to make a design sufficiently flexible. Grossmann et al. (1983) established the optimization strategies for flexible chemical process, which were emphasized on two major areas: optimal design with a fixed degree of flexibility and design with optimal degree of flexibility. Saboo and Morari (1985) proposed a resilience index to characterize the largest total uncertainty, which a HEN can tolerate while remaining feasible. Kotjabasakis and Linnhoff (1986) introduced a procedure for the design of flexible HENs, which was aimed at establishing the trade-off between energy, capital, and flexibility. Picon Nunez and Polley (1995) developed an approach to solve the flexibility problems of heat recovery networks by the use of network simulation and suggested the modification strategies and steps. Picon Nunez and Polley (2000) also incorporated a consideration of operability into the design of multistream heat exchangers (MHE). The operability characteristics of MHEs are considered satisfying if the steady-state response to disturbances of specified magnitude are within acceptable bounds. Roetzel and Luo (2001) proposed a set of sensitivity matrices of one-dimensional MHEs when study their flexibility, which were applied to determine the regulations of mass flow rates of fluids in MHEs. Of all these works, the efforts of Papalexandri and Pistikopoulos (1994a,b) to incorporate flexibility and controllability considerations simultaneously in the synthesis and retrofit design of HEN are the most systematic works. They gave a systematic framework for the synthesis of cost optimal heat exchanger networks that satisfy a set of controllability criteria. Based on a hyperstructure network representation, dynamics are explored via the introduction of an analytical model for dynamic gains and time delays in a heat exchanger network. Input-output pairing criteria are explicitly included within a mixedinteger nonlinear programming (MINLP) formulation for the synthesis of HEN. Network configurations can be obtained featuring minimum total annualized cost controllability characteristics, which may form the basis for a successful control scheme.

9.2.1 Flexibility and structural controllability considerations In the process of synthesis and retrofit design of HEN, the structural controllability targets can be expressed with different kinds of criteria. Papalexandri and Pistikopoulos (1994a) defined the target as the total disturbance rejection with respect to the specified controlled variables and disturbances. To consider flexibility and structural controllability, we denote the set of the uncertain variables of the network by u, the specified range of

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Design and operation of heat exchangers and their networks

uncertainty by Ru, and the set of the design variables by d. Let x represent the state variables that describe the network operation; c represent controlling variables corresponding to the degrees of freedom at the design stage; z represent the controlled variables, which is a subset of x; zref is the set value vector of z; and the deviation Δz ¼ z  zref. The synthesis and retrofit problems with structural controllability target were formulated by Papalexandri and Pistikopoulos (1994a) as follows, respectively. Synthesis problem of HEN: min TACsynthesis

d , x, c

(9.25)

hðd, x, c, uÞ ¼ 0

s:t:

gðd, x, c, uÞ  0 8u 2 Ru 9cjfjΔzj  Δzmax g Retrofit problem of HEN: min TACretrofit

Δd, x, c

(9.26)

hðd, x, c, uÞ ¼ 0

s:t:

gðd, x, c, uÞ  0 8u 2 Ru 9cjfjΔzj  Δzmax g where h(d, x, c, u) is the set of equality constraints and g(d, x, c, u) the set of inequality constraints. Halemane and Grossmann (1983) showed that the flexibility requirements in the previous problems are equivalent to the inequality constraint that involves a max-min-max problem:   max min max gj d, x, c, u  0 (9.27) u2Ru

c

j2Jg

s:t: hðd, x, c, uÞ ¼ 0 where Jg is the index set for the inequalities g. For the controlled variables z to be undisturbed from the disturbance inputs u, it suffices GR½Mðx, c, z, uÞ  AR½Mðx, c, z, uÞ  1

(9.28)

where M(x, c, z, u) denotes the structural matrix of the network, GR is the generic rank of M, and AR is the number of its active rows.

Optimal control process of heat exchanger networks

441

The generic rank of M is determined by the following optimization problem (Georgiou and Floudas, 1989): m X n X yij max i, j i¼1 j¼1

(9.29)

m X yij  vi ¼ 0 ði ¼ 1, 2, ⋯, nÞ s:t: j¼1

n X

yij  wj i¼1 n X

 0 ðj ¼ 1, 2, ⋯, mÞ

vi ¼ GR

i¼1 m X

wj ¼ GR

j¼1

in which vi is a binary variable corresponding to a row and denoting whether the row is active (when vi ¼ 1) or redundant (when vi ¼ 0), wj is a binary variable corresponding to a column and denoting whether the column is active (when wj ¼ 1) or redundant (when wj ¼ 0), and yij is a binary variable indicating that the variable of the jth column is an output variable of the ith row if yij ¼ 1. yij is set to zero when the corresponding elements of M are zero. The synthesis and retrofit problems can be equivalently written as min TAC

Δd, x, c

 s:t: s:t:

(9.30) 

max min max gj ðd, x, c, uÞ  0 u2Ru

c

j2Jg

hðd, x, c, uÞ ¼ 0 GR½Mðx, c, z, uÞ  AR½Mðx, c, z, uÞ  1

9.2.2 Dynamics and control structure considerations Dynamic controllability deals with the maintenance of the target output parameters upon short-term deviations of inlet parameters or operation transitions from one operating point to another. The dynamic controllability depends on dynamic characteristics of the HEN. Papalexandri and Pistikopoulos (1994b) used a multiperiod hyperstructure to present structural, operational, and control alternatives in a HEN network. The multiperiod hyperstructure for a stream k is shown in Fig. 9.2.

442

Design and operation of heat exchangers and their networks

ak,k¢,s yk,k¢,s wk,k¢,s

zk,k¢,s

xk,k²,k¢,s¢,s bk,s

Stream k wk,k²,s¢

xk,k¢,k²,s,s¢

zk,k²,s¢

yk,k²,s¢ ak,k²,s¢

Fig. 9.2 Hyperstructure network representation.

Based on this hyperstructure, possible control alternatives can be considered. Then, the synthesis/retrofit problem can be formulated as a mixed-integer nonlinear programming (MINLP) problem. As is shown in Fig. 9.2, the manipulating splitters (cycles) are located at the inlet of a stream with the stream splittings wk,k0 ,s and wk,k00 ,s0 and the overall bypass bk,s, prior to an exchanger with the exchanger bypass ak,k0 ,s or ak,k00 ,s0 , and at the outlet of a stream from an exchanger with the multibypass xk,k00 ,k0 ,s0 ,s or xk,k0 ,k00 ,s,s0 . For selecting the control scheme, Papalexandri and Pistikopoulos (1994b) introduced the pairing binary variables to denote that a specified output is controlled through a split fraction or through utility loads. The pairing variables are interconnected, so that for each controlled outlet temperature one manipulating variables is selected, and each manipulating variable is assigned to at most one controlled output. The dynamic controllability considerations can be represented by introducing time delays in the network. Papalexandri and Pistikopoulos (1994b) developed a variable delay matrix, where the time response of an output to an input is given as a function of the network structure and design parameters to be determined, such as stream flows, heat exchanger areas, and pipe characteristics. For a given HEN, the time-response paths of an output to an input can be easily determined. As is shown in Fig. 9.3, the network in this case is represented by a directed graph with variable nodes (splitters, mixers, and exchanger units) connected by arrow lines. For example, the heat exchanger bypasses are represented by “hex” and “cex” for hot and cold streams, respectively, and “e” denotes the time delay through the exchanger wall due to the wall capacitance. The time delay of each arrow line can be

Optimal control process of heat exchanger networks

443

Fig. 9.3 Time delay graph.

determined with the lump parameter model suggested by Mathisen and Morari (1994). Then, the time delay matrix of the HEN and its control structure can be expressed as a function of the network operating point. To formulate the controllability requirements, Papalexandri and Pistikopoulos (1994b) proposed two time delay–based criteria. The one is for dynamic decoupling, which is expressed as   τij  τij0  1  mij M1 + α1  0 8i, j, j0 (9.31) where τij is the response time of a controlled variable i to its pairing manipulating variable j; τij0 is the response time of the controlled variable i to other manipulating variables j0 (j0 6¼ j); mij is the pairing variable that defines the control of i through j; and α1 is a positive variable, which shall be maximized so that, for a control pair ij, the response time of the controlled variable i to the manipulating input j is as small as possible compared with the response time to any other manipulating input j0 . M1 is a large positive number, M1 > αu1 where αu1 is a valid upper bound of α1, so that the criterion, Eq. (9.31) becomes a redundant constraint when mij ¼ 0. The other criterion is the time effective control expressed by   τij  τik  1  mij M2 + α2  0 8i, j, k (9.32) Differing from Eq. (9.31), here, k indicates a disturbance input, and the index “2” refers to the parameters for the time effective control. This criterion implies that the response time of the control variable i to the

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Design and operation of heat exchangers and their networks

manipulating input j shall be as small as possible compared with the response time to any disturbance input k. This criterion is in fact for a feedforward control; however, since it will yield a quick response of a controlled output to the corresponding manipulating input, we can use it also for a feedback control.

9.2.3 Minimal interaction requirements An effective control structure also requires a minimal interaction among the manipulating variables. We hope that a controlled variable i shall be more sensitive to its pairing manipulating variable j and less sensitive to other manipulating variables k (k 6¼ j). Papalexandri and Pistikopoulos (1994b) further introduced a minimal interaction criterion into the synthesis model as follows:   Gik  Gij  1  mij M3 + α3  0 8i, j, k ðk 6¼ jÞ (9.33) where i indicates a controlled output ϕi, j indicates a pairing manipulating input cj, k indicates a nonpairing manipulating input ck (k 6¼ j), and Gik is the gain of the controlled output ϕi to the manipulating input ck, which is defined by   ∂ϕ  Gik ¼  i  (9.34) ∂ck The synthesis and retrofit design with controllability requirements can be expressed by Eq. (9.30) with additional constraints (9.31)–(9.33).

9.3 Operation of heat exchanger networks under uncertainty A HEN is usually designed at given fixed operating conditions. However, there may exist deviations of some inlet parameters of process streams. A HEN designed with the methods described in the previous sections should have a satisfied operability and controllability to shift between different operation states, since they have already been considered in the synthesis/ retrofit process. In this section, we will discuss its online optimization or the control between two steady states. To find methods for the operation of HENs, Mathisen et al. (1992) investigated bypass selection for control of HEN, without considering the utility consumption. Glemmestad et al. (1997) proposed a method for optimal operation of HENs. Their method uses steady-state optimization carried out online with regular time intervals.

Optimal control process of heat exchanger networks

445

The results of this optimization are then implemented by specifying the optimal value of some variables. The control objective between different steady states is to regulate other inlet parameters to maintain the target output parameters of process streams (usually the output temperatures of the process streams) either exactly at the nominal operation conditions or as close as possible to the nominal operation conditions. According to the degrees of freedom of HEN to achieve the control objectives, the operation changes of HENs can be divided into three cases: the number of regulatable parameters is greater than, or equal to, or less than the number of target output temperatures. If the deviations are within the feasible region of a HEN, there may exist only one set, or more than one sets of regulation solutions, to maintain the former target output temperatures. For the case of more than one solutions, the optimal solution should be determined, which yields an optimization problem. For a HEN, the regulation of stream flow rates is easy to be realized. So we focus on the tasks for maintaining the stream temperatures. To determine a needed regulation, the relation between temperature deviations and the change of inlet parameters should be obtained first, which involves the rating process of a HEN. To develop the relations to calculate temperature deviations, take a two-stream heat exchanger with bypass control scheme as an example, as is shown in Fig. 9.4, in which t0 h and t00 h (or t0 c and t00 c) are the temperatures of the hot (or cold) stream coming to the splitter and leaving from the mixer, t0 E,h and t00 E,h (or t0 E,c and t00 E,c) the temperatures of the hot

chC˙ h t¢h C˙ h

t¢E,h

t²E,h

(1–ch)C˙ h

t²c

t²h

(1–cc)C˙c t²E,c

t¢E,c Heat exchanger

Fig. 9.4 Heat exchanger with bypass control scheme.

C˙ c t¢c

ccC˙ c

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Design and operation of heat exchangers and their networks

(or cold) stream entering and exiting the heat exchanger, C_ h and C_ c the thermal capacity rates of the hot and cold streams, and ch and cc the bypass fractions for hot and cold streams, respectively. For the steady-state operation condition, if the heat loss is negligible, we have t0 E,h ¼ t0 h and t0 E,c ¼ t0 c. To present whether a bypass control exists, we can further introduce a binary variable y associated with the bypass fraction c. If there exists a bypass control, y ¼ 1, otherwise y ¼ 0. The exit stream temperatures of the ith heat exchanger can then be obtained from Eqs. (9.13), (9.14) as th00, i ¼ ½1  ð1  yh, i ch, i Þεi th0 , i + ð1  yh, i ch, i Þεi tc0 , i

(9.35)

tc00, i ¼ ð1  yc, i cc, i ÞRi εi th0 , i + ½1  ð1  yc, i cc, i ÞRi εi tc0 , i

(9.36)

where NTUi and Ri are calculated with thermal capacity rates through the exchanger: NTUi ¼ Ri ¼

ki Ai ð1  yh, i ch, i ÞC_ h, i

ð1  yh, i ch, i ÞC_ h, i ð1  yc, i cc, i ÞC_ c, i

(9.37) (9.38)

The dimensionless temperature change of the hot stream is calculated with Eqs. (9.17)–(9.22). For a general HEN composed of NE heat exchangers including heaters and coolers for heating Nc cold process streams and cooling Nh hot process streams, we use the following matrix relationship derived from Eqs. (9.35), (9.36) to obtain the outlet temperatures of the heat exchangers: AT00 ¼ T0

(9.39)

where T contains the nodal temperatures of the HEN and A is a coefficient matrix obtained according to Eqs. (9.35), (9.36) for each heat exchanger of the network. For a HEN running at the nominal operation point, Eq. (9.39) is expressed as AN T00N ¼ T0N

(9.40)

If there exist parameter deviations of the process streams from their nominal values, the deviations of all nodal temperatures θ of the HEN can be evaluated by   θ ¼ T00  T00N ¼ ðAN + ΔAÞ1 T0N + ΔT0  T00N (9.41)

Optimal control process of heat exchanger networks

447

Eq. (9.41) demonstrates that the deviations of the controlled output variables Δz  θ will be affected by the deviations in the thermal capacity rates of the process streams C_ h, j ( j ¼ 1, 2, …, Nh) and C_ c, k (k ¼ 1, 2, …, Nc) and the manipulation of the bypass fractions ch,i and cc,i (i ¼ 1, 2, …, NE). If the HEN is flexible, for a given u 2 Ru, an optimal control vector c 2 Rc (Rc ¼ {0  ch, i  1, 0  cc, i  1 jyi ¼ 1; i ¼ 1, 2, …, NE}) can be found such that the total utility cost reaches the minimum, n X o X (9.42) min C ðx, c, uÞ ¼ min a QHU ðx, c, uÞ + b QCU ðx, c, uÞ c2Rc

c2Rc

s:t: Δzðx, c, uÞ ¼ 0 hðx, c, uÞ ¼ 0 gðx, c, uÞ  0

9.4 The dynamic control of heat exchanger networks Operation analysis (static optimal) of a given HEN gives us the information on which variables (or which bypasses) should be manipulated and how much should be manipulated upon disturbances in order to maintain the target temperatures so that the utility consumption at the same time is minimized. The dynamic control analysis and control strategy design will decide how to manipulate those selected bypasses within a limited time to recover the nominal operation outputs, namely, how to arrange the temporary manipulations and how fast the change should be carried out. The dynamic control process is depended on the choice of control system and its strategy.

9.4.1 Basic concepts of the model predictive control Feedback and feedforward are two types of control schemes for systems that react automatically to the dynamical disturbance inputs. The control of any dynamic system, including heat exchangers and HENs, should be executed on the basis of automatic control principle. Traditional automatic control is based on PID feedback method, by which the feedback is the sum of proportional plus integral plus derivative. As is shown in Fig. 9.5 as an example, for controlling the outlet temperature of the hot stream at the set value, t00 h,set, the deviation Δz ¼ z  zref between the measured outlet temperature t00 h and its target temperature t00 h,set are sent to the feedback controller. According to Δz and its variation history, the controller sends a manipulating signal y to adjust the bypass fraction so that the deviation Δz will approach zero again. Like all other PID feedback control systems, during the control process, the

448

Design and operation of heat exchangers and their networks

y

Feedback controller Z

chC˙ h

t²h,set

t¢h C˙ h

Zref

t¢E,h

t²E,h

t²h

(1–ch)C˙ h C˙ c t²c

t¢c Heat exchanger

Fig. 9.5 Feedback control system.

manipulation of input parameters is carried out only after the deviation of the controlled output variables has been detected; therefore lag error and sometime undesired fluctuations and/or instability cannot be avoided. A feedforward control method was developed to overcome these shortcomings. In a feedforward control system, the controller senses the deviation of inlet parameters and give the corresponding adjustments based on the predication of dynamic response of the system, as is shown in Fig. 9.6. For the traditional feedforward control, dynamic behavior of the system is determined by analytical methods or experimental methods, which may be expressed in differential equation form or transfer function form, and they can be converted each other by mathematical methods. In order to improve the control quality of feedforward control in its application in a HEN, to predict the transient responses of the HEN more accurately becomes necessary for the proper manipulation. Guan et al. (2004) used a mathematical model based on a distributed parameter approach to predict the dynamic behaviors of a plate-fin HE and the change of output parameters caused by certain disturbances can be calculated. The dynamic analysis model is embedded in the controller. When a disturbance of input parameter occurs, or the system should be switched from one state to a new operation state, the predictive model in the controller will predict the response of output parameters based on the sense of input parameter change, and then the controller instructs the manipulation of bypass valves; thus, the change of output

Optimal control process of heat exchanger networks

449

Z DZ

Ztar chC˙ h c

t²h,set

C˙ h

Model predictive controller

t¢h

t¢E,h

t²E,h

(1–ch)C˙ h t¢c

u1 u2

u3 u4

t²h

C˙ c

t²c Heat exchanger

Fig. 9.6 Model predictive feedforward control system.

parameter is compensated or the target change is carried out. The control process is essentially a compensation process. Model predictive feedforward control is an open-loop control, but different from the common open-loop control, where the adjustments of input parameters are determined according to the relationships of the transfer functions (matrix) by applying the inverse Laplace transform to it, here, the model predictive controller manipulates according to the solution of transient responses of inlet disturbances; thus, better qualities (absolute errors and speed limits) are ensured. Especially when a distributed parameter approach is adopted to set up the model, the transient responses of the whole system upon disturbances can be predicted, including the inner parameters like temperature distribution inside the heat exchanger or HEN. This is a great help to improve the control quality, for example, when a bypass valve manipulates upon a certain disturbance to maintain the target outputs or to switch to a new operation state, because of the inertia of the system, quick manipulation is expected to eliminate the delay of the outputs, which could result in overadjustment. Small overmanipulation of outputs can usually be tolerated, but when the overmanipulation of outputs appears, unacceptable overshot of inner parameters such as unexpected high temperature could happen. By distributed parameter approach, inner parameters can be calculated at the same time so that these can be chosen as special constraints. The other features of distributed parameter model predictive control are the following: (1) It can be designed for multivariable processes; based on the solution of dynamics of multistream heat exchangers and their networks, a set of

450

Design and operation of heat exchangers and their networks

responses corresponding to different input disturbances can be given at the same time, while the traditional open-loop control can only compute the corrective action to one input variable. Therefore, this accurate model predictive feedforward control system can be used for multistream heat exchanger control, where the traditional open-loop control is difficult to be implemented. (2) As previously mentioned, the accurate model predictive control can choose the optimal control scheme from many choices under different constraints, which is impossible by the traditional open-loop control. The implementation of distributed parameter model predictive feedforward control is a reverse question of dynamic solution obtained. The controller should give the corrective instruction based on the response solution upon the changes of objective input parameters to adjust the inputs of assistant fluid, namely, the changes of inputs should be given based on the response solution, but the solution obtaining calculation is to find the output response based on the information of input changes; the former is much difficult to be carried out. Thus, two steps are needed in the control execution: firstly, the solution obtaining calculation is implemented according to the change of objective input parameter, and then, the adjustment of auxiliary input is obtained by iterative calculations. The model predictive feedforward control is a big improvement to traditional feedback control, but it still needs improvement. Because of the accuracy of the model adopted and the inertia of the system, it may not give the complete compensation when disturbances happened, and for no signal of output is feedback to the controller, the possible error cannot be adjusted. If the output error is feedback and the controller will act based on the integrated response, the system is a “true” model predictive control according to the control theory. The model predictive control system can be schematically shown in Fig. 9.7, where the feedback is included, but it is different from traditional feedback and feedforward system in that it is not the output, but the error of output compensation is feedback. Since Richalet et al. (1978), the predictive control method was developed by many researches (Clarke et al., 1987a,b; Soeterboek, 1990; Richalet, 1993). Some applications of model predictive control had been given for heat exchanger control (Hecker et al., 1997). Several books and thousands of papers about the model predictive control have been published. Intensive reviews of the developments in model predictive control and their applications were given by Garriga and Soroush (2010), Tran et al. (2014), Vazquez et al. (2014),

451

Optimal control process of heat exchanger networks

Model parameter modification

Z Zpredicted chC˙ h c C˙ h

t²h,set

Ztar

Model predictive controller

t²E,h

t¢E,h

t¢h

(1–ch)C˙ h t¢c

u1 u2

u3 u4

t²h

C˙ c

t²c Heat exchanger

Fig. 9.7 Model predictive control system with feedback compensation.

Lio et al. (2014), Ellis et al. (2014), Karamanakos et al. (2014), Lee (2011), and Mayne (2014).

9.4.2 Control system for heat exchanger networks Boyaci et al. (1996) suggested a procedure for HEN control based on repeated steady-state optimization. When the HEN experiences a disturbance u, the static optimization is to find a set of bypass fractions c that will minimize the total deviations of the controlled variables jjΔz jj subject to the given constraints such that the HEN remains feasible: min kzðx, c, uÞ  ztar k

c2 Rc

(9.43)

s:t: hðx, c, uÞ ¼ 0 gðx, c, uÞ  0 It should be noted that the objective function used here is not the usual minimum utility consumption. The task of the dynamic control part is to determine how to apply in time the set of optimal bypass fractions c so that the HEN’s dynamic response in reaching the final steady-state feasible operating point is acceptable. The first control strategy is the optimal open-loop control logic, where there is no feedback of the state variables x. According to this scheme, the disturbance u applied to the HEN are measured at discrete time intervals. These measurements are fed to the optimizer that performs the static optimization mentioned previously by referring to the steady-state algebraic model of

452

Design and operation of heat exchangers and their networks

the HEN. The optimizer finds the optimal values of the bypass fraction, copt, such that the target values of the controlled variables are satisfied. Assuming that there is no mismatch between the static model of the optimizer and the process (HEN), the remaining task, at this point, is to find how to implement these optimal bypass fractions as a function of time. Since there is no offset between the predictions of the algebraic and dynamic models at steady state, the bypasses can be opened up to their optimal values either instantaneously or as a function of time, for example, as a ramp starting from their nominal values cN toward their optimal values copt within the time interval Δτ, as is represented by Eq. (9.44) for the ith bypass manipulation: (   τ cN, i + copt, i  cN, i , τ < Δτi (9.44) ci ðτÞ ¼ Δτi copt, i , τ  Δτi The choice of the time interval Δτi depends on the dynamic characteristics of the HEN and can be determined by experience, or by solving an additional optimization problem: ð∞ min kz½x, cðτ=ΔτÞ, u, τ  ztar kdτ (9.45) Δτ

0

s:t: hðx, c, uÞ ¼ 0 gðx, c, uÞ  0 For the optimal closed-loop control logic of HENs, there is a feedback of the state variable xp pairing to the corresponding bypasses, xp  x. The function proposed for the implementation of the optimal controls is suggested as   xp ðτÞ  xp, N cðτÞ ¼ cN + copt  cN xp, opt  xp, N

(9.46)

Sun et al. (2018) proposed a methodology for two-stage coordination of bypass control and economic optimization. Based on this methodology, they developed a one-step coordination between the bypass control and economic optimization. Two kinds of control manipulations are defined in this method: bypass control cb and economic optimization control ce. In the two-stage coordination control, firstly, the fraction of bypass is adjusted quickly for meeting the control requirement: " Nc  #  b  X cb, ini, j  cb, j  min + w kzðx, cb , ce, ini , uÞ  ztar k (9.47) cb 2Rcb cb, ini, j j¼1

Optimal control process of heat exchanger networks

453

s:t: hðx, cb , ce, ini , uÞ ¼ 0 gðx, cb , ce, ini , uÞ  0 where cb,ini is the initial bypass fraction, ce,ini the initial valve opening for the economic operation, cb the new bypass fraction, and w the weighting factor. Then, the economic optimization described as follows is solved: min C ðx, cb , ce , uÞ ¼

ce 2Rce

h X i X min a QHU ðx, cb , ce , uÞ + b QCU ðx, cb , ce , uÞ

ce 2Rce

(9.48)

s:t: zðx, cb , ce , uÞ  ztar ¼ 0 hðx, cb , ce , uÞ ¼ 0 gðx, cb , ce , uÞ  0 In the one-step coordination control, the two controlling variables are manipulated simultaneously by solving the dynamic optimization problem with an optimization algorithm with pattern search (Hooke and Jeeves, 1961) combining a penalty function for the constraints. According to Sun et al. (2018), we can express the optimization problem as h min Ck ðx, cb , ce , uÞ + wΔCk ðx, cb , ce , uÞ cb 2Rcb , ce 2 Rce X + φi jhi ðx, cb , ce , u, τk + 1 Þj +

X

i

 i φj min gj ðx, cb , ce , u, τk + 1 Þ, 0

(9.49)

j

where φi and φj are the penalty factors for different constraints and w is the weighting factor. The utility costs Ck and ΔCk are calculated with Eqs. (9.50)–(9.53): ð τk + 1 h X 1 Ck ðx, cb , ce , uÞ ¼ a QHU ðx, cb , ce , u, τÞ τ k + 1  τk τk i X (9.50) + b QCU ðx, cb , ce , u, τÞ dτ ΔCk ðx, cb , ce , uÞ ¼ Ck, max ðx, cb , ce , uÞ  Ck, min ðx, uÞ h X _ Ck, max ðx, cb , ce , uÞ ¼ max a QHU ðx, cb , ce ,u, τÞ τ2½τk , ∞Þ i X _ + b QCU ðx, cb , ce ,u, τÞ

(9.51)

(9.52)

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Design and operation of heat exchangers and their networks

n X a QHU ½x, cb , ce , uðτk + 1 Þ min cb 2Rcb , ce 2 Rce o X + b QCU ½x, cb , ce , uðτk + 1 Þ

Ck, min ðx, uÞ ¼

in which

uðτÞ ¼ _

uðτÞ, τ  τk + 1 uðτk + 1 Þ, τ > τk + 1

(9.53)

(9.54)

The time step τk represents the control horizontal, 0 ¼ τ0 < τ1 < τ2 <… < τk < τk+1. Ck is the mean utility cost in the kth time interval [τk, τk+1], Ck,min is the steady-state minimum utility cost under the optimal controlling variables cb and ce against the disturbance u at τk+1, and Ck, max is the possible maximum utility cost in the time range of [τk, ∞) due to the disturbance expressed by Eq. (9.54).

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