Synthesis of Optimum Controllable Heat Exchanger Networks using ...

Synthesis of Optimum Controllable Heat Exchanger Networks using Genetic Algorithms B. Tech. Project submitted in partial fulfillment of the requirements for the degree of Bachelor of Technology in Chemical Engineering by Ashish Pattekar under the guidance of Prof. R. D. Gudi & Prof. U. V. Shenoy

Department of Chemical Engineering Indian Institute of Technology Bombay April 13, 1999

ACKNOWLEDGEMENT

I would like to express my sincere gratitude to my guide Prof. R. D. Gudi and my co-guide Prof. U. V. Shenoy for their continuous support and valued guidance throughout the course of this project and in the preparation of this report.

Ashish V. Pattekar April 13, 1999

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Abstract The last few decades have seen considerable research into the synthesis of optimal heat exchanger networks (HENs). Process integration has paved the way for designs that are optimal from the point of view of energy recovery. Subsequent optimization of net exchanger area in such designs results in a network that is optimal from the point of view of utility consumption and exchanger area. Of late, attempts have been made to incorporate controllability considerations into the synthesis of optimal HENs. The modern HEN synthesis problem can therefore be stated as the simultaneous optimization of a HEN design from the point of view of Energy, Area, and Control. Structural or static controllability has been considered in the design optimization of HENs in the past. The effect of disturbances in case of operating HENs, however, remains to be considered. This is one of the important motivations behind this project. The final aim is to achieve simultaneous optimization of the HEN design over all the aspects stated above. A novel search procedure, Genetic Algorithms (GAs), is proposed for the synthesis and optimal design of HENs. The search domain being huge and the behavior of the objective function being rather noisy, the search procedure used should be such that it is not affected by the presence of a large number of local optima. The main reason for using this search method is its relative immunity to the nature of the objective function. A basic GA has been developed for analysis. Various modifications in this basic GA have been made to suit the present problem. Details of the implementation of various design criteria for the search are discussed along with the results from the preliminary application procedure. Methods for further improving the search procedure are suggested.

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Contents

Acknowledgement

1

Abstract

2

Nomenclature

7

1 Introduction and Literature Survey

1

1.1

Optimization Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Maximum Energy Recovery . . . . . . . . . . . . . . . . . . .

2

1.1.2

Minimum Exchanger Area . . . . . . . . . . . . . . . . . . . .

2

1.1.3

Minimum Net Operating Costs . . . . . . . . . . . . . . . . .

2

1.2

A Good Search Procedure . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Objective and Scope of the work . . . . . . . . . . . . . . . . . . . . .

4

1.4

Report Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2 Control System Selection and Comparison 2.1

2.2

Multivariate Interacting Systems: RGA Approach . . . . . . . . . . .

5

2.1.1

Interpreting the RGA Elements . . . . . . . . . . . . . . . . .

6

The Niederlinski Index . . . . . . . . . . . . . . . . . . . . . . . . . .

8

3 Controllability Issues 3.1

5

10

Propagation of Disturbances and Manipulations . . . . . . . . . . . .

11

3.1.1

11

The Steady-State Behavior . . . . . . . . . . . . . . . . . . . . 3

3.1.2

Effects of temperature variations . . . . . . . . . . . . . . . .

12

3.1.3

Structural Singularities . . . . . . . . . . . . . . . . . . . . . .

13

3.2

Right Half Plane Zeroes . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.3

Dynamic Controllability . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.3.1

The Disturbance Cost Method . . . . . . . . . . . . . . . . . .

14

3.3.2

A Quantitative Controllability Index . . . . . . . . . . . . . .

15

4 Search using Genetic Algorithms

19

4.1

A very basic GA . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.2

Data Representation . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4.3

Selection Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

4.4

The Genetic Operators . . . . . . . . . . . . . . . . . . . . . . . . . .

21

4.5

Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

4.6

Search Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

5 GA implementation

24

5.1

Basic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

5.2

Data Representation . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

5.3

Preliminary Implementation Details . . . . . . . . . . . . . . . . . . .

25

5.4

Application Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

5.5

A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.6

Application Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.7

An Improved Strategy . . . . . . . . . . . . . . . . . . . . . . . . . .

30

6 Further Implementations 6.1

32

An Area Optimization Algorithm . . . . . . . . . . . . . . . . . . . .

33

6.1.1

The Conventional GA . . . . . . . . . . . . . . . . . . . . . .

33

6.1.2

The va06a/ad subroutine . . . . . . . . . . . . . . . . . . . . .

33

6.1.3

The Improved Genetic Algorithm (IGA) . . . . . . . . . . . .

34

4

6.2

Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . .

35

6.3

Complex Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

6.4

Optimization for Controllability . . . . . . . . . . . . . . . . . . . . .

40

7 Conclusion and guidelines for further work

42

Reference

44

5

List of Figures 3.1

Controllability of Heat Exchanger Networks. . . . . . . . . . . . . . .

17

3.2

Overall HEN synthesis and optimization . . . . . . . . . . . . . . . .

18

4.1

A simple 10-bit binary string crossover. . . . . . . . . . . . . . . . . .

20

5.1

Representation of Heat Exchanger Networks. . . . . . . . . . . . . . .

25

5.2

Application of crossover operator to HEN . . . . . . . . . . . . . . . .

25

5.3

Cross Match (1) not represented suitably . . . . . . . . . . . . . . . .

27

5.4

Data and optimal solution for the 4SP1 problem (Papoulias and Grossmann, 1983). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.5

Optimum design reported by the GA due to a Systematic Random search 29

5.6

An improved Strategy . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

Numerical example (Linnhoff and Ahmad, 1989) (TAC = $7.025 ×

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107 /yr.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

6.2

Optimal design reported by the GA-IGA for the problem in figure 6.1

35

6.3

Search History for the structure optimization program(Main GA) . .

36

6.4

Search History for the area optimization program (IGA) . . . . . . .

37

6.5

Nine Stream HEN design problem (Linnhoff and Ahmad, 1989) (TAC

6.6

= $2.89 × 106 /yr.) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

HEN Design obtained from IGA. (TAC = $2.91 × 106 /yr.) . . . . . .

39

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Nomenclature Symbol

Description

A

Exchanger Area

GT G¯(s) G¯

Transfer function matrix for inlet temperatures System transfer function for manipulated variables

m ¯ (s)

Manipulated variables

NT U

Number of transfer units

P

Thermal Effectiveness

Rh

Heat capacity flow rate ratio

T

Temperature

U U¯(s)

Overall Heat Transfer co-efficient

U¯d(s)

Disturbances

w Y¯(s) ¯ λ

Heat Capacity flow rate

∆Thx

Overall temperature driving force

d(s)

System transfer function for disturbances

Manipulated variables

Matrix of manipulated variables Relative Gain Array

Subscripts and Superscripts i

Inlet

o

Outlet

h

Hot Stream

c

Cold Stream

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Chapter 1 Introduction and Literature Survey The basic optimal Heat Exchanger Network (HEN) synthesis problem can be stated as follows: “Design a network of Heat Exchangers using available hot and cold utility if necessary capable of changing the temperatures of various hot and cold process streams to desired levels with the minimum capital and operating costs.” While the capital costs are mainly determined by the total number of heat exchangers and the net exchanger area, the operating costs depend on a variety of factors, chief among them being the utility consumption. As a result, the optimum design of HENs has traditionally revolved around minimizing the exchanger area and utility consumption. In this chapter the different criteria to be considered in the optimization of HENs are discussed and a review of the literature available on the subject is presented. The optimization method to be used in the implementation is also discussed.

1.1

Optimization Criteria

The different criteria used in the optimization of Heat Exchanger Networks are based on their contribution to the capital or operating costs. Some important criteria to be considered in this project are discussed below.

1

1.1.1

Maximum Energy Recovery

When a number of hot and cold streams are already present in the process, it is worthwhile to use the energy from the hot streams to heat up the cold streams as far as possible as this will result in a lower net utility consumption. This process of using heat from process streams themselves is called energy recovery. Maximum Energy Recovery (MemR) designs (Linnhoff and Hindmarsh, 1983) are those in which maximum possible energy required to heat the cold streams is derived from the hot streams themselves. Consequently MeR designs are designs with minimal external utility consumption. This method of exchanging energy amongst the process streams for lower net utility consumption is known as Process Integration. Various methods for determining the MeR designs are available in literature. ‘Pinch Technology’ (Linnhoff and Hindmarsh, 1983) is one such classic method used to determine MeR designs. ‘The dual-temperature design method’ (Trivedi et al., 1989) is another MeR design technique which allows for more flexibility at the second step of capital cost optimization.

1.1.2

Minimum Exchanger Area

After the various possible designs for minimum energy are obtained, the network is then checked for minimum area by simply selecting the design that requires minimum total exchanger area from the possible MeR networks. Various guidelines are available in literature for obtaining designs with minimum number of units and minimum area. The papers by Linnhoff (1994) and Galli and Cerdia (1998) provide brief reviews of the state of the art of this technology. The book by Shenoy (1995) contains an exhaustive summary of the literature on this topic.

1.1.3

Minimum Net Operating Costs

Operating costs do not depend on the utility consumption alone, though it does form an important part the total operational expenses. With more and more designs incorporating process integration, it has been found that the performance of the HENs under actual operation might suffer from certain inherent design limitations (Mathisen, 1994) that translate into a higher net operational cost. Certain designs that are indeed very good from Energy and Area point of view might be very difficult 2

to control in case of flow rate or temperature fluctuations in the process streams. Such designs will have very high net operating costs and are best avoided. It is therefore important that the overall operating costs for any HEN design be found after considering a number of factors, such as Operability (Gundersen et al.,1996), Static and Dynamic Controllability (Lin, 1974; Mathisen, 1994; Papalexandri and Pistikopoulos, 1994), and Disturbance Resiliency (Lewin, 1996). Only after a design has been synthesized with due consideration to all the above factors can it be claimed to be totally optimal.

1.2

A Good Search Procedure

In order to find an optimum design from the set of all possible designs, a robust search procedure is necessary. The objective function in this case, which is the net capital and operating cost, can be expected to exhibit a number of local minima due to the very nature of the problem. A good search procedure should not be affected by the presence of such local minima, or the noisy nature of the objective function as such. Another important characteristic of a good search procedure is flexibility. Although optimal designs from separate considerations for Energy Recovery or Area can be obtained using a number of methods available in theory, such an approach may lead to missing out those designs that might be slightly below optimal from Energy point of view but very good for control, thus leading to a net lower operating cost. This has been the main consideration in choosing Genetic Algorithms (Holland, 1975; Goldberg, 1989) as a search procedure for optimal synthesis of HENs in this project. With the option of either using theoretical considerations to narrow down the search domain or using a pure search over the entire domain, the search procedure can be made flexible enough to cover either the entire domain or certain desirable sub-domains on the lines of traditional search methods. Detailed considerations for the implementation of the selection criteria mentioned above are discussed in the following chapters. The actual application of the GA procedure along with the approach for encoding the designs into suitable form is also presented.

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1.3

Objective and Scope of the work

The objective of this work has been to achieve simultaneous HEN optimization from the point of view of overall capital and operating costs which include all the aspects mentioned above. Attempts have previously been made to consider controllability in HEN design. However, most of the work in this direction (Papalexandri and Pistikopoulos, 1994; Huang and Fan, 1992) is directed in and evolutionary fashion involving different criteria at different stages in the optimization process by progressively cutting down the search space. Though this indeed leads to very good designs, the overall optimality of the final design may become questionable at times (Galli and Cerdia, 1998). It is therefore proposed to attempt a simultaneous optimization of all the criteria as far as possible. This is one of the main reasons behind choosing Genetic Algorithms as a search procedure as it allows considerable flexibility in the general optimization philosophy during the final implementation (Goldberg, 1989). The consideration of disturbance effects is another motivation behind this project. Static or structural controllability and operability has been considered for the design of HENs in the past (Huang and Fan, 1992; Mathisen, 1994; Papalexandri and Pistikopoulos, 1994), but the effect of disturbances remains to be considered in detail. It is proposed to incorporate this criterion through the implementation of the Disturbance Cost method (Lewin, 1996) and the Quantitative Controllability Index based on the consideration of surge volumes (Zheng and Mahajanam, 1999) in the final optimization procedure using genetic algorithms.

1.4

Report Outline

The Relative Gain Array, a simple method for selection of feedback control configuration in case of multivariate systems will be used in the analysis. A review of this method is presented in Chapter 2. Various controllability issues, the disturbance cost method and the Quantitative Controllability Index are discussed in detail in Chapter 3 of this report with a summary of the relevant literature. An overview of Genetic Algorithms is provided in Chapter 4. Chapters 5 and 6 contain the major contribution of this work involving the implementation of the application of genetic algorithms to HEN optimization. The details of the implementation along with the application results are presented in these Chapters. 4

Chapter 2 Control System Selection and Comparison The Heat Exchanger Network is a multivariate system, with a number of manipulated and controlled variables. Loop pairing of manipulated and controlled variables for feedback control requires optimal selection of the manipulated variables corresponding to the controlled variables. Consequently, the designs themselves can be compared with each other after deciding individual control strategies. In this chapter, the use of the Relative Gain Array (RGA) (Bristol, 1966) for control configuration selection is discussed along with the approach for comparison of control configurations in different systems. Implementation for selection of designs that are optimal from control point of view is then considered.

2.1

Multivariate Interacting Systems: RGA Approach

Multiple single loop feedback control of multivariate systems involves the selection of pairs of manipulated inputs and controlled outputs to form feedback loops. In the presence of interaction among different loops, the control action initiated in one loop may cause changes in other loops that are best avoided. These changes may in turn initiate control action in these loops leading to retaliationary action against the main loop. The aim of feedback loop selection is then to minimize all interactions as far

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as possible. The RGA is a simple loop selection procedure that leads to minimally interacting designs. Consider a system for which the controlled variables Y¯(s) are given as a function of manipulated variables m¯(s) by the equation

Y¯(s) = G¯(s) m¯(s)

(2.1)

¯ such that Then the Relative Gain Array for this system is defined as the matrix λ



λij =  δy 

δyi δmj

 All loops open

i

δmj

(2.2)

All loops closed except f or the mj loop

or,

!

λij =

open − loop gain f or loop i under the control of mj closed − loop gain

(2.3)

Interpretation of the RGA elements is discussed below.

2.1.1

Interpreting the RGA Elements

In order to interpret the RGA elements, it is convenient to classify the values into ranges as discussed below. • λij = 1 : This indicates that the loop i will not be subject to retaliationary action from other loops when they are closed, so mj can be used to control yi without interference from other loops. It is therefore recommended that pairing yi with mj will be ideal.

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• λij = 0 : This indicates that the open loop gain between yi and mj is zero. It is therefore recommended not to pair yi with mj , though it will be advantageous to pair mj with some other output, as one can be sure that at least yi will be immune to interaction from this loop. • 0 < λij < 1 : This indicates that the open loop gain between yi and mj is smaller than the closed loop gain. It may therefore be concluded that the loops are definitely interacting, but the retaliationary effect from the other loops is in the same direction as the main effect of mj on yi . It is recommended that such pairing be avoided if λij < 0.5, as this indicates that the retaliationary action from the other loops is actually greater than that of the main loop. • λij > 1 : This means that the interactions from the other loops are opposite in direction but smaller in magnitude than effect of the main loop. It is recommended that such pairing be avoided if the value of λij is much greater than unity. • λij < 0 : This means that the closed loop gain is opposite in sign to the open loop gain, therefore the retaliationary effect from the other loops is not only in opposite direction to the main loop, but also of a higher magnitude. This may lead to the system becoming unstable as any control action will only result in moving the system farther away from the desired operating point. It is therefore recommended that yi not be paired with mj . It is clear that the closer λij is to unity, the better it is to control the ith controlled output using the j th manipulated input. Therefore the best control configuration would be one in which the diagonal elements of the RGA are closest to unity, and the rest are closest to zero, as this means that the interactions initiated on other loops by any control action in any particular loop are the minimum possible. In practice, while applying the RGA method for loop pairing, it is not necessary to calculate the RGA from first principles. Special matrix methods are available in literature for quick calculation of the RGA.

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2.2

The Niederlinski Index

In addition to the guidelines given above, the Niederlinski theorem (Niederlinski, 1971) gives a reasonably accurate method of determining the best control configuration for a system. Consider the n × n multi-variable system whose input and output variables have been paired as follows: y1 − u1 , y2 − u2 , ... , yn − un , resulting in a transfer function of the form:

Y¯(s) = G¯(s) U¯(s)

(2.4)

Further, let each element of G¯(s) be rational and open loop stable. Also let n individual feedback controllers be designed with integral action such that each one of the n feedback control loops is stable when rest of the n − 1 loops are open. Then, under closed loop condition of all the n loops, the system will be unstable for all possible values of controller parameters if the Niederlinski index (N ) defined below is negative: i.e,

|G¯(0) |