optimization of a liquefaction plant using genetic algorithms

Optimization of a liquefaction plant usinggenetic algorithms

G. Cammarata, A. Fichera *, D. Guglielmino

Istituto di Fisica Tecnica, UniversitaÁ di Catania, Viale A. Doria 6, 95125 Catania, Italy

Received 23 March 2000; received in revised form 6 July 2000; accepted 9 July 2000

Abstract

This paper presents an optimization methodology for liquefaction/refrigeration systems inthe cryogenic ®eld. The Figure of Merit has been chosen as the evaluation index, and geneticalgorithms as evaluation criteria. This methodology has been applied to an existing helium

liquefaction system that works according to a Collins cycle. The results show the possibility ofadjusting some of the thermal-pressure variables for the system in order to improve the Figureof Merit. # 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Cryogenics; Genetic algorithms; Optimization methodology

1. Introduction

A cryogenic system is a device used to obtain temperatures in the range 0±150 K.Such devices have applications in the medical and biological ®elds, in aerospaceengineering, in superconductivity, and in the air separation and gas liquefaction(helium, hydrogen) industries. The Claude cycle is the basis of most helium andhydrogen refrigeration and liquefaction systems. It may be regarded as a combina-tion of the Linde and Brayton cycles [1].With regard to optimization criteria, there are several approaches described in the

literature. It is possible to carry out an optimization using the ®rst law of thermo-dynamics, the second law, a second law analysis, the entropy generation concept,and the exergy. In [2], the entropic optimization of helium refrigeration/liquefaction

Applied Energy 68 (2001) 19±29

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* Corresponding author. Tel.: +390-95-337-994; fax: +390-95-337-994.

E-mail address: a®[email protected] (A. Fichera).

cycles was developed; the characteristic system variables were identi®ed to describethe system and a parametric analysis of the irreversibility of each component wasperformed. In [3], an analysis of a helium refrigerator (Claude type) was discussed;the trend of the ratio of refrigeration work to refrigeration heat was determined by®xing the number of expanders and speci®c independent variables. In [4], a generalmethod for optimising the ideal Claude cycle for large size refrigerators with multi-ple expansion engines was tested on two di�erent arrangements. In [5], a sensitivityanalysis with respect to some fundamental parameters was presented and the

Nomenclature

ef liquid exergyFOM ®gure of merithf liquid enthalpyhIN-Comp enthalpy at the compressor inletk ratio of speci®c heats(=1.667 for helium gas)m:

Expÿi ¯ow to ith expanderm:f liquid helium ¯ow

ns number of populationpIN-Comp pressure at the compressor inletpIN-Exp-i inlet pressure to ith expanderpm mutation probabilitypOUT-Comp pressure at the compressor outletpOUT-Exp-i outlet pressure to ith expanderR ideal gas constantsf liquid entropysIN-Comp entropy at the compressor inletT0 reference temperaturetc crossover indexTIN-Comp temperature at the compressor inletTINExp-i inlet temperature to the ith expandertm mutation indexTOUTExp-i outlet temperature to the ith expanderTf ¯uid temperatureW:

Comp compression powerW:

Expÿi power of ith expanderW:

IDEAL ideal workWR ®tnessW:

REAL actual work�Comp compressor e�ciency�Ex energetic e�ciency of the plant�Expÿi e�ciency of ith expander

20 G. Cammarata et al. / Applied Energy 68 (2001) 19±29

relationships between thermodynamic parameters of a helium liquefaction andrefrigeration plant were de®ned. In [6], an analysis of frequently used helium refrig-eration cycles was described, summarising some major steps in e�ciency improvement.From the variety of approaches cited, one may conclude that a general optimiza-

tion method for di�erent cryogenic system con®gurations does not exist. In fact, thecriteria described above are usually applied to large-scale systems and, as analysedin [7], comparisons between them cannot be easily performed because of di�erencesin both the working conditions and in the analysed typologies. Moreover, the opti-misations are typically based on only a few parameters, so they do not allow one todetermine an optimum con®guration that takes all of the parameters into accountsimultaneously. In fact, in the literature, there are optimisation methods that con-sider only a couple of parameters, but the optimisation problem is more complexand requires the simultaneous resolution of many variables. To solve this problem itis possible to use a procedure of complex combinatory calculations, which allow oneto determine the optimum set of values for all the variables.In the present paper, a simple methodology is proposed; it is applicable to any

kind of liquefaction plant, working with any type of cycle. This methodology isaimed at determination of the optimum value of the Figure of Merit (FOM) for theplant. The optimization criteria are obtained by the heuristic procedures of GeneticAlgorithms (GAs), which represent a valid tool for the resolution of complex com-binatory problems. Using this method, it has been possible to extrapolate from aninitial population, the con®gurations corresponding to the best FOMs, by means ofa probabilistic natural selection process. This paper speci®cally analyses, as anapplication, an existing and operating helium liquefaction plant.

2. Proposed methodology

The proposed optimization method, referred to plants for cryogenic gas liquefac-tion based on Claude cycle, is applicable to plants working on any other cycles.For the system's performance evaluation the Figure of Merit (FOM) has been

considered as an index parameter. The general formulation of FOM [7] referred topower terms and, considering n expanders, has been modi®ed according to thescheme of Fig. 1, becoming:

FOM �

ÿW: IDEAL

m:f

!ÿW: REAL

m:f

! � To ��sINÿComp ÿ sf� ÿ �hIN-Comp ÿ hf�W:

Comp ÿPn1

W:

Expÿi

� �� 1m:f

�1�

The numerator coincides with the liquid's speci®c exergy ef :

ef � hf ÿ hIN-Comp

ÿ �ÿ To sf ÿ sIN-Comp

ÿ � �2�

G. Cammarata et al. / Applied Energy 68 (2001) 19±29 21

Fig. 1. Diagram of a Claude cycle plant.


Therefore, the FOM coincides with the exergetic e�ciency of the plant �ex, since itis calculated as the ratio between the speci®c exergy of liquid ef and the real speci®cliquefaction work:

FOM � ÿW: IDEAL

m:f

!=ÿW: REAL

m:f

!� ef=

ÿW: REAL

m:f

!� �Ex �3�

It must be underlined that the identity between FOM and exergetic e�ciency ise�ective when the expansion work is really recovered. Formulation of the FOM by(1) can be kept for the purposes of energetic optimization, as shown below.By considering the equations of compression and expansion power, it is possible

to write the FOM as:

FOM � T0 � sIN-Comp ÿ sfÿ �ÿ hIN-Comp ÿ hf

ÿ �

1

m:f�

1

�Comp�m: Comp �R�T0 �ln pOUT

pIN

� �Comp|��{z��}

isothermal compression work

ÿXn1

�Expÿi �m: Expÿi �R�TIN-Expÿi � k

kÿ 1� 1ÿ pOUT

pIN

� �kÿ1k

Expÿi

" #|��{z��}

adiabaitc expansion work

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;�4�

Previous equations express that the FOM depends both on temperature andpressure conditions of the nodal points of the plant and on its con®guration, i.e. onthe number of expanders and on their layout.Fixing some working parameters of the plant [e.g. those indicated in Eq. (5)], the

Figure of Merit is a function depending on the following thermo-pressure variables:

FOM � FOM pIN-Comp;TIN-Comp;Tf;m:; pOUT-Comp;TIN-Expÿi;TOUT-Expÿi

� ��5�

or, considering the numerator of the FOM's expression as equal to a constant, theoptimization is brought back to the minimisation of the function placed in thedenominator of expression (4).Since the FOM is a function of n-variables, its optimization requires a search of a

conditioned minimum in the space of Rn dimension. For this purpose, genetic algo-rithms have been used [8, 9]. The calculation code ``Genesis 5.0'' developed by JohnJ. Grefenstette [10] using programming language C has been chosen for the solutionof the problem herein examined.


Genesis 5.0 code allows one to choose the characteristic parameters of geneticalgorithm, speci®cally:

. the number of population fellows, ns;

. the crossover index, the fraction of population to which the crossover geneticoperator tC is applied;

. the mutation index, the percentage of bits subject to mutation, tm.These parameters have been ®xed to the default values for a ®rst attempt and then

modi®ed to test the algorithm outputs for varying internal factors.

3. Examined problem

Fig. 2 shows the entropy versus temperature diagram for the reference cycle hereinused. The characteristic values and the helium ¯ows for each component are alsodisplayed for the maximum production rate of the liquid helium ¯ow.

Fig. 2. Reference cycle of liquefaction system in the entropy versus temperature diagram.


The lique®er consists of two turbo-expanders, six heat exchangers and one JouleThompson expansion valve. It works as a modi®ed Collins cycle. In fact, the twoexpanders are placed in series on the same line. This means that the output ¯owfrom the ®rst expander (position 11 in Figs. 1 and 2) is sent to the inlet of the secondexpander, after a transfer in an exchanger (position 12); the output of the lastexpander is then sent to the low pressure line of the machine (position 13). The restof the high pressure helium is expanded by the Joule Thompson valve up to state 10,to obtain the liquid helium ¯ow at state 9.Referring to Fig. 2, the terms in Eq. (4) can be expressed as:

. the dividend (the work really necessary to a complete the liquefaction of unit¯ow of the ¯uid in the cycle):

ÿW: IDEAL

m:f� T20 � s20 ÿ s9� � ÿ h20 ÿ h9� � �6�

. the divisor (the work necessary for liquefaction of the unit ¯ow using the realcycle):

ÿW: REAL

m:f�

W:

Comp ÿPn1

W:

Expÿi

m:f

�7�

where:

W:

Comp � 1

�Comp�m: 20 �R�T20 �ln p1

p20�8�

Xn1

W:

Expÿi �W:

Expÿ1 �W:

Expÿ2 �9�

being:

W:

Expÿ1 � �Expÿ1 �m: 4 �R�T4 � k

kÿ 1� 1 ÿ p11

p1

� �kÿ1k

!�10�

W:

Expÿ2 � �Expÿ2 �m: 12 �R�T12 � k

kÿ 1� 1 ÿ p20

p12

� �kÿ1k

!�11�

In the analysis, ®xed e�ciencies and no losses due to friction were assumed. Fixingthe liquid ¯ow to be obtained (mf), the temperature of liquefaction (state point 9, T9)and the pressure and temperature at the inlet of the compression step (state 20, T20,


p20), the ideal liquefaction work is determined and is constant. The denominator, orFOM, by the previous expressions is a function of the draining pressure of thecompressor (state 1, p1), of the inlet temperature at the second expander (state 12,p12) and of the inlet temperature and draining pressure of the ®rst turbine (respec-tively state 4, p4 � p1, and 11, p11 � p12):

FOM � FOM p�20;T� 20;T� 9; p4;T4; p12;T12

� ��12�

The minimum of the liquefaction real work, that means the maximum of the FOMfor the plant, depends on the values of four parameters associated with the nodes``4'' and ``12'', or on the thermo-pressure conditions of the input and output of thetwo expanders. In the con®guration shown in the T-S diagram the FOM is equal to:

FOM �

ÿW: IDEAL

m:f

!ÿW: REAL

m:f

! � T20 ��s20 ÿ s9� ÿ �h20 ÿ h9�W:

Comp ÿP21

W:

Expÿi

� �� 1m:f

� 0:0593 � 5:93% �13�

The liquid helium properties were obtained from McCarty [11].

4. Results

As described, the numerator of the FOM is constant, so to maximize the ratio it isenough to minimize the denominator. Particularly, since m

:f is a ®xed value,

obtained from the operating conditions of the plant, it is possible to maximize FOMby minimizing the following objective function:

WR � 1

�Comp�m: 20 �R�T20 �ln p1

p20

� �ÿm:4

k

kÿ 1�R

� 1ÿ p12p4

� �kÿ1k

" #�T4 ��Expÿ1 ÿ 1ÿ p20

p12

� �kÿ1k

" #�T12 ��Expÿ2

( )�14�

The range of values, in which it is possible to obtain the solution, has been deter-mined by the following considerations:

. T4 falls in the range 70±100 K. The lower limit of this range has been deter-mined considering that pre-refrigeration, using liquid air, has not been used(this type of refrigeration would allow one to obtain helium at the output ofthe ®rst exchanger). The higher limit of the range depends on the low e�-ciencies required for the two exchangers placed before section ``4'';


. p1 is acceptable if contained in the range 1.1±2.0 MPa. The lower value of therange allows one to achieve the quality of the mixture in the two-phase zone ofthe helium diagram so, assuring the production of helium without modifyingthe performance of the part of the plant before the lamination valve. Thehigher value of the range is linked to excessive heating to which the gas issubject during the compression step;

. p12 is to be found in the range 0.3±0.7 MPa, where these values have been chosenin order not to eliminate the small enthalpic drop in one of the two turbines;

. T12 may vary over the range 15±25 K, in order not to excessively penalise thee�ciency of the exchanger placed between sections 11 and 12.

Once de®ned, the ranges of values of the independent variables (inlet temperaturesand pressures of the two expanders), and the values of the parameters of the geneticalgorithm, were varied to achieve the optimum value of ®tness or to verify, to goodapproximation, that the value obtained is an absolute and not a local minimum inthe de®ned ®eld of variables. The results reported in Tables 1 and 2 show that theminimum of the WR reached during the simulation is not a local value but anabsolute minimum. In fact, ®xing the number of generations and varying the char-acteristic values of the Genetic Algorithm (tm, tc, nP) by discrete steps, the corre-sponding values of the WR are very close to the minimum reached previously. Inparticular, the minimum value of ®tness (WR) and the corresponding characteristicvalues of the Genetic Algorithm are:

Table 1

Values of ®tness, WR, varying the crossover index

Generations=200, Index tm � 0:001

Crossover index tc No. of members of population

10 30 50

0.6 100,007 99,9838 99,9769

0.7 100,007 100,007 99,9773

0.8 100,1315 99,9869 99,9777

Table 2

Values of ®tness, WR, varying the mutation index

Generations=200, Index tc � 0:6

Mutation index tm No. of members of population

10 30 50

0.003 99,9876 99,994 99,996

0.009 100,019 99,978 99,994

0.018 99,988 99,9769 99,986


WR � 99:9769 kW� � tc � 0:6; tm � 0:001 and ns � 50

Once the values of the parameters corresponding to the minimum of the objectivefunction have been ®xed, an additional veri®cation consists in obtaining an asymp-totic law of the ®tness trend versus the number of generations. As shown in Fig. 3,the value of ®tness, WR, becomes constant after 200 generations, con®rming, again,that the determined value for WR is an absolute minimum. The values of the inde-pendent variables and FOM corresponding to the minimum of the objective func-tion are reported in Table 3. Comparison with the value previously obtained showsan improvement of 10% in the FOM.

5. Conclusions

A system for helium liquefaction has been analysed. To this end, an optimizationmethodology based on genetic algorithms has been de®ned.Versatility of this meth-odology allows one to optimise the system by choosing a suitable number of inde-pendent variables that are su�cient to characterise the plant.In the application considered, the optimization of a liquid helium production

system working under maximum ¯ow conditions was carried out; under these con-ditions an improvement of 10% in the FOM was achieved.

Table 3

Parameter values corresponding to the minimum of the objective function

p4 (MPa) T4 (K) p12 (MPa) T12 (K) WR (kW) FOM

New values 1.1 78.72 0.401 24.4 99.97 0.066

Old values 1.52 81.20 0.66 17.5 111.4 0.0593

Fig. 3. Values of objective function WR versus number of generations.


References

[1] Hands BA, Alii. Cryogenic Engineering. London: Academic Press, 1986.

[2] Hubbell H, Toscano WM. Thermodynamic optimization of helium liquefaction cycles. Advances In

Cryogenic Engineering 1980;25:551±62.

[3] Khalil A, McIntosh GE. Thermodynamic optimization study of the helium multi-engine Claude

refrigeration cycle. Advances In Cryogenic Engineering 1977;23:431±7.

[4] Hilal MA. Optimization of helium refrigerators and lique®ers for large superconducting systems.

Cryogenics 1979;7:415±20.

[5] Narinsky GB, Temirova VV, Chernokov LV. In¯uence of thermodynamic parameters on main

characteristics of helium liquefaction and refrigeration plant. Cryogenics 1995;35:483±7.

[6] Ziegler B, Quack H. Helium refrigeration at 40 percent e�ciency? Advances in Cryogenic Engineer-

ing 1992;37:645±51.

[7] Timmerhaus KD, Flynn TM. Cryogenic process engineering (the international cryogenic monograph

series). New York: Plenum Press, 1989.

[8] Goldberg DE. Genetic algorithms in search optimization and machine learning. USA: Addison-

Wesley, 1989.

[9] Davis L. Handbook of genetic algorithms. New York: Van Nostrand Reinhold, 1990.

[10] Grefenstette JJ, Davis L, Cerys D. GENESIS and OOGA: two genetic algorithm systems. TPS, 1991.

[11] McCarty RD. Thermodynamic properties of helium from 2 to 1500 K with pressures to 100 MPa,

Software of the O�ce of Standard Reference Data Ð National Bureau of Standards (now the

National Institute of Standards and Technology, NIST), Washington, DC, USA.


optimization of a liquefaction plant using genetic algorithms

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