Ind. Eng. Chem. Res. 1993,32, 2633-2635 2633

PROCESS DESIGN AND CONTROL

Optimization of Heat Exchanger Networks

Rein Luus Department of Chemical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4

In the optimization of heat exchanger networks, an easy and reliable method of obtaining the global optimum has been overlooked by recent researchers. It is shown that the direct search optimization procedure using randomly chosen test points and region contraction as developed by Luus and Jaakola over 20 years ago is ideally suited for these types of problems. The global optimum can be obtained an order of magnitude faster than the recently proposed method by Quesada and Grossmann.

Introduction

Recently Quesada and Grossmann (1993) presented a method for optimizing heat exchanger networks to avoid getting a local optimum. The difficulty of obtaining the global optimum for highly nonlinear problems that possess numerous local optima was highlighted by Luus et al. (1992) in considering the optimization of a bifunctional catalyst blend problem. Although iterative dynamic programming yielded the global optimum without much difficulty, the use of sequential quadratic programming with 100 starting points yielded numerous local optima, none of which was the global optimum. For the same system Storey (1992) found that other optimization procedures, including simulated annealing, failed to locate the global optimum. However, Bojkov et al. (1992) showed that the direct search optimization method of Luus and Jaakola (1973) yielded the globaloptimum with reasonable reliability for a wide range of starting conditions.

Since the direct search optimization procedure is easy to use, especially on personal computers, ita suitability to heat exchanger networks needs to be determined. The purpose of this paper is to show that, through rearrange- ment of the equations describing the heat exchanger networks, the optimization problem becomes very simple.

Optimization Problem

If the equations describing a heat exchanger network are rearranged so that variables can be solved serially, then direct search optimization can be applied very effectively. This was illustrated by Jaakola and Luus (1974) in the optimization of a complex chemical process. If the equations cannot be arranged conveniently, a two- pass method as suggested by Luus (1974) and illustrated by the optimization of a multistage recycle system optimization by Luus (1975) may be used. Therefore, let us rearrange the equations given by Quesada and Gross- mann (1993) to set up the optimization problem.

The problem is to minimize

Q Q Q3 Q4 C = 2 7 d + 7200-1 + 240-- + 90+ (1) AT, AT2 AT3 AT4

subject to the equality constraints

Q2 = 1000 - Q1

T, = 398 + Q,/3.125

Q3 = 5.555(575 - TI) T3 = 365 + Q3/4.545

Q4 = 3.125(718 - TJ T4 = 358 + Qd3.571

ATl = 0.5(T1 - 305)

AT2 = 0.5(T2 - 302)

AT3 = 0.5(T1 - T3 + 210)

AT4 = o.5(T2 - T4 + 360)

(3)

(4)

(5) (6)

(7) (8 )

(9)

(10)

(11)

(12)

T, 1405 (13) T, I 405 (14)

T, I 5 7 5 (15) T, I 718 (16) T3 1 365 (17)

T4 1 358 (18) ATi I 5 i = 1, ..., 4 (19)

Qi 1 0 i = 1, ..., 4 (20)

and the inequality constraints

It is noted that if TI is chosen as a free variable, then eqs 2-12 yield all the variables required for evaluating the cost function in eq 1. As was shown by Quesada and Grossmann (19931, when C is plotted against 81, the function possesses a local minimum at around Q1 = 168.4 kW and a global minimum at Q1 = 978.1 kW. These correspond to TI = 425.3 K and TI = 571.1 K, respectively.

Results and Discussion For computations a 486133 personal computer was used.

For the optimization procedure of Luus and Jaakola (19731, the built-in random number generator in Microsoft Fortran was used. The listing of the optimization program is given in Table I.

Since the range for TI is between 405 and 575, a natural midpoint for the region for TI is 490 and the initial region

0ass-5885/93/2632-2633$04.00/0 0 1993 American Chemical Society

2634 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993

Table I. Listing of the Optimization Program

35

37

52

43 30 94

73

51

95 100

197

IMPLICIT DOUBLE PRECISION(A-H,O-21 PI\RAMETERlN~1,LIM-2O,NIT-2O1,NPTS-lO,RED-O.95DOl DIMENSION X(4) ,REGl4) ,XP(4) ,XSl4) RERc'4 RN OPEN ( W I T - 3 , FILE*'XEATl .OUT' ) NO - 0 ~. LW-LIM

Do 94 J-1,NPTS Do 52 I-1.N CALL RANWMIRN) RAN I RN - 0.5 IF(J.EQ.1) RAN - O.OD0 XLII - XP(I1 t RAN*REGlI) IF(X(1) .LT.405.DO) Xll) - 405.ODO IF(X(1) .GT.575.DO) X(1) .I 575.D0 T1 = XI11 Q1 I 5.555DO*IT1-395.ODO) Q2 - 1000.DO - Q1 IF(Q2.LT.O.ODO) GO TO 30 T2 I Q2/3.125DO + 398.0DO IF(T2.LT.405.DO) GO TO 30 IF(TZ.GT.718.DO) GO TO 30 Q3 I 5.555DO*l575.0DO - T1) T3 I 365.D0 + Q3/4.545DO IF(T3.LT.365.DO) GO TO 30 Q4 - 3.125DO+l718.DO - T2) T4 I 358.D0 + Q4/3.571DO IF(T4.LT.358.DO) GO TO 30 DT1 I 0.5DO*lT1-305.DO) IF(DTl.LT.5.ODO) GO TO 30 DT2 = 0.5DO*(T2-302.DO) IF(DT2.LT.S.ODOl GO TO 30 DT3 - O.SDO*ITl-T3 + 210.ODO) IF(DT3.LT.5.0) GO TO 30 DT4 I 0.5DO*(T2-T4+360.ODO) IF(DT4 LT.S.DO) GO TO 30 C = Z.;D3*Ql/DTl + 7.2D3*Q2/DT2 + 2.4D2*Q3/DT3 + 9.OD2*Q4/DT4 IF(C.GT.TEST) GO TO 30 NO - NO + 1 TEST - C F M - c W 43 I-l,N XSII) I X(1) CONTINUE CONTINUE 1FlLW.LT.LIM) GO TO 51 WRITE(3.73) IT,NO, IXSII) ,I-l,N) ,FM FORMATIlX,217,5F12.3) LW.0 LW I LW+1 Do 95 I-1.N REQII) -REG(Il*RBD XP(1) - XSII)

STOP END

size of 170 is used to cover the entire range. The region reduction factor is chosen to be 0.95, since this value appears to be a good one to use for a variety of problems (Wang and Luus, 1978; Spaans and Luus, 1992). Although here we have a one-dimensional search, the program has been written so that the four-dimensional problem in example 7 of Quesada and Grossmann (1993) can be run with only minor changes.

The first run using nine random points and the midpoint of the region of each iteration, which corresponds to the given program listing, yielded rapid convergence to the global optimum of C = 36 163 as is shown in Table 11. The optimum is reached already after 141 iterations. It is interesting to note that, in spite of the large number of function evaluations, the CPU time for 201 iterations was only 0.13 s.

By using 4 random points per iteration rather than 9, the convergence rate is slightly slower, requiring 161 iterations to reach 36 163, as is shown in Table 111. The total CPU time for 201 iterations, however, is reduced to 0.06 s. It is noted that the cost function is very sensitive to TI. When TI is changed from 571.080 to 571.066, the cost function changes from 36 163 to 36 167.

As is shown in Table IV, the use of only two random points per iteration yielded the local optimum of 45 686 with TI = 425.3. This may explain why simulated

Table 11. Program Output with Nine Random Points per Iteration (CPU Time = 0.13 8 )

iter no. 1 21 41 61 81 101 121 141 161 181 201

NFE 5 17 30 50 70 76 93 114 130 149 149

Ti C 557.309 570.994 571.042 571.042 571.042 571.070 571.077 571.078 571.080 571.080 571.080

39 599.979 36 187.530 36 173.787 36 173.787 36 173.787 36 165.902 36 163.842 36 163.518 36 163.018 36 162.986 36 162.986

Table 111. Program Output with Four Random Points per Iteration (CPU Time = 0.06 8 )

iter no. NFE Ti C 1 21 41 61 81 101 121 141 161 181 201

4 24 25 25 25 25 26 27 28 42 62

542.482 570.862 571.066 571.066 571.066 571.066 571.073 571.076 571.080 571.080 571.080

42 252.501 36 225.351 36 167.113 36 167.113 36 167.113 36 167.113 36 164.880 36 164.254 36 163.088 36 162.992 36 162.992

Table IV. Program Output with Two Random Points per Iteration (CPU Time = 0.03 8 )

iter no. 1 21 41 61 81 101 121 141 161 181 201

NFE 3 26 46 49 59 80 101 121 141 161 181

Ti 437.861 425.920 4 2 5.9 2 0 4 2 5.4 3 3 425.243 425.308 425.312 425.312 425.312 425.312 425.312

C 45 879.545 45 686.924 45 686.924 45 686.234 45 686.215 45 686.205 45 686.205 45 686.205 45 686.205 45 686.205 45 686.205

annealing or adaptive random search may have difficulty in obtaining the global optimum from an arbitrary starting condition. Generally, even for a one-dimensional search, as this one, the use of fewer than nine points per iteration is not advised, especially if existence of numerous local optima is suspected. For this particular problem, taking between 4 and 80 random points per iteration for 201 iterations takes less than 1 s of CPU time on a personal computer and yields the global optimum with a very high probability.

Concluding Remarks Simple optimization procedures that give reliable results

are available for use on personal computers. In this particular instance the global optimum can be obtained without any difficulty whatsoever. What makes the direct search optimization procedure of Luus and Jaakola (1973) so versatile is the ease with which the number of test points used in each iteration and the reduction factor can be changed. The effect of these parameters in the optimi- zation of a more complex system has been shown by Spaans and Luus (1992). The heat exchanger network optimi- zation problem is relatively simple, and it appears that the Luus and Jaakola optimization procedure works exceedingly well. Since there are no auxiliary functions to be evaluated, the method is very easy to use and is computationally fast.

Literature Cited Bojkov, B.; Hansel, R.; Luus, R. Application of direct search

optimization to high-dimensional optimal control problems. Proceedings of the 42nd Canndian Chemical Engineering Con- ference, Toronto, Ontario, Oct 18-21,1992; University of Toronto Press: Toronto, ON, pp 285-286.

Jaakola, T. H. I.; Luus, R. A note on the application of nonlinear programming to chemical-process optimization. Oper. Res. 1974, 22,415-417.

Luus, R. Two-pass method for handling difficult equality constraints in optimization. AZChE J. 1974,20, 608-610.

Luus, R. Optimization of multistage recycle systems by direct search. Can. J. Chem. Eng. 1975,53, 217-220.

Luus, R.; Jaakola, T. H. I. Optimization by direct search and systematic reduction of the size of search region. AZChE J. 1973, 19,760-766.

Luus, R.; Dittrich, J.; Keil, F. J. Multiplicity of solutions in the optimization of a bifunctional catalyst blend in a tubular reactor. Can. J. Chem. Eng. 1992, 70, 780-785.

Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2635

Quesada, I.; Grossmann, 1. E. Global optimization algorithm for heat exchanger networks. Znd. Eng. Chem. Res. 1993,32,487-499.

Spaans, R.; Luus, R. Importance of search-domain reduction in random optimization. J. Optim. Theory Appl. 1992,74636-638.

Storey, C. Global optimization in chemical engineering. Proceedings of the 42nd Canadian Chemical Engineering Conference, Toronto, Ontario, Oct 18-21,1992; University of Toronto Press: Toronto,

Wang, B. C.; Luua, R. Reliability of optimization procedures for ON, pp 13-14.

obtaining global optimum. AIChE J. 1978,24,619424.

Received for review April 26, 1993 Revised manuscript received August 23, 1993

Accepted August 30, 1993.

* Abstract published in Advance ACS Abstracts, October 15, 1993.