Optimization of a liquefaction plant using genetic algorithms

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<ul><li><p>Optimization of a liquefaction plant usinggenetic algorithms</p><p>G. Cammarata, A. Fichera *, D. Guglielmino</p><p>Istituto di Fisica Tecnica, Universita` di Catania, Viale A. Doria 6, 95125 Catania, Italy</p><p>Received 23 March 2000; received in revised form 6 July 2000; accepted 9 July 2000</p><p>Abstract</p><p>This paper presents an optimization methodology for liquefaction/refrigeration systems inthe cryogenic field. 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</p><p>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.</p><p>Keywords: Cryogenics; Genetic algorithms; Optimization methodology</p><p>1. Introduction</p><p>A cryogenic system is a device used to obtain temperatures in the range 0150 K.Such devices have applications in the medical and biological fields, 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</p><p>literature. It is possible to carry out an optimization using the first 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</p><p>Applied Energy 68 (2001) 1929</p><p>www.elsevier.com/locate/apenergy</p><p>0306-2619/01/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.PI I : S0306-2619(00 )00041 -6</p><p>* Corresponding author. Tel.: +390-95-337-994; fax: +390-95-337-994.</p><p>E-mail address: afichera@ift.ing.unict.it (A. Fichera).</p></li><li><p>cycles was developed; the characteristic system variables were identified 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 byfixing the number of expanders and specific 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 dierent arrangements. In [5], a sensitivityanalysis with respect to some fundamental parameters was presented and the</p><p>Nomenclature</p><p>ef liquid exergyFOM figure of merithf liquid enthalpyhIN-Comp enthalpy at the compressor inletk ratio of specific heats(=1.667 for helium gas)m:</p><p>Expi flow to ith expanderm:f liquid helium flow</p><p>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 fluid temperatureW:</p><p>Comp compression powerW:</p><p>Expi power of ith expanderW:</p><p>IDEAL ideal workWR fitnessW:</p><p>REAL actual workComp compressor eciencyEx energetic eciency of the plantExpi eciency of ith expander</p><p>20 G. Cammarata et al. / Applied Energy 68 (2001) 1929</p></li><li><p>relationships between thermodynamic parameters of a helium liquefaction andrefrigeration plant were defined. In [6], an analysis of frequently used helium refrig-eration cycles was described, summarising some major steps in eciency improvement.From the variety of approaches cited, one may conclude that a general optimiza-</p><p>tion method for dierent cryogenic system configurations 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 dierencesin 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 configuration 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</p><p>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 configurations corresponding to the best FOMs, by means ofa probabilistic natural selection process. This paper specifically analyses, as anapplication, an existing and operating helium liquefaction plant.</p><p>2. Proposed methodology</p><p>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 systems performance evaluation the Figure of Merit (FOM) has been</p><p>considered as an index parameter. The general formulation of FOM [7] referred topower terms and, considering n expanders, has been modified according to thescheme of Fig. 1, becoming:</p><p>FOM </p><p>W: IDEALm:f</p><p> !W: REAL</p><p>m:f</p><p> ! To sINComp sf hIN-Comp hfW:</p><p>Comp Pn1</p><p>W:</p><p>Expi</p><p> 1m:f</p><p>1</p><p>The numerator coincides with the liquids specific exergy ef :</p><p>ef hf hIN-Comp To sf sIN-Comp 2</p><p>G. Cammarata et al. / Applied Energy 68 (2001) 1929 21</p></li><li><p>Fig. 1. Diagram of a Claude cycle plant.</p><p>22 G. Cammarata et al. / Applied Energy 68 (2001) 1929</p></li><li><p>Therefore, the FOM coincides with the exergetic eciency of the plant ex, since itis calculated as the ratio between the specific exergy of liquid ef and the real specificliquefaction work:</p><p>FOM W:</p><p>IDEAL</p><p>m:f</p><p> !=W: REAL</p><p>m:f</p><p> ! ef= W</p><p>:REAL</p><p>m:f</p><p> ! Ex 3</p><p>It must be underlined that the identity between FOM and exergetic eciency iseective 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</p><p>to write the FOM as:</p><p>FOM T0 sIN-Comp sf hIN-Comp hf </p><p>1</p><p>m:f</p><p>1</p><p>Compm: Comp RT0 ln pOUT</p><p>pIN</p><p> Comp|{z}</p><p>isothermal compression work</p><p>Xn1</p><p>Expi m: Expi RTIN-Expi kk 1 1</p><p>pOUTpIN</p><p> k1k</p><p>Expi</p><p>" #|{z}</p><p>adiabaitc expansion work</p><p>8&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;:</p><p>9&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;=&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;;4</p><p>Previous equations express that the FOM depends both on temperature andpressure conditions of the nodal points of the plant and on its configuration, 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</p><p>Figure of Merit is a function depending on the following thermo-pressure variables:</p><p>FOM FOM pIN-Comp;TIN-Comp;Tf;m: ; pOUT-Comp;TIN-Expi;TOUT-Expi </p><p>5</p><p>or, considering the numerator of the FOMs 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</p><p>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.</p><p>G. Cammarata et al. / Applied Energy 68 (2001) 1929 23</p></li><li><p>Genesis 5.0 code allows one to choose the characteristic parameters of geneticalgorithm, specifically:</p><p>. the number of population fellows, ns;</p><p>. the crossover index, the fraction of population to which the crossover geneticoperator tC is applied;</p><p>. the mutation index, the percentage of bits subject to mutation, tm.These parameters have been fixed to the default values for a first attempt and then</p><p>modified to test the algorithm outputs for varying internal factors.</p><p>3. Examined problem</p><p>Fig. 2 shows the entropy versus temperature diagram for the reference cycle hereinused. The characteristic values and the helium flows for each component are alsodisplayed for the maximum production rate of the liquid helium flow.</p><p>Fig. 2. Reference cycle of liquefaction system in the entropy versus temperature diagram.</p><p>24 G. Cammarata et al. / Applied Energy 68 (2001) 1929</p></li><li><p>The liquefier consists of two turbo-expanders, six heat exchangers and one JouleThompson expansion valve. It works as a modified Collins cycle. In fact, the twoexpanders are placed in series on the same line. This means that the output flowfrom the first 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 flow at state 9.Referring to Fig. 2, the terms in Eq. (4) can be expressed as:</p><p>. the dividend (the work really necessary to a complete the liquefaction of unitflow of the fluid in the cycle):</p><p>W: IDEALm:f T20 s20 s9 h20 h9 6</p><p>. the divisor (the work necessary for liquefaction of the unit flow using the realcycle):</p><p>W: REALm:f</p><p>W:</p><p>Comp Pn1</p><p>W:</p><p>Expi</p><p>m:f</p><p>7</p><p>where:</p><p>W:</p><p>Comp 1Comp</p><p>m: 20 RT20 ln p1p20</p><p>8</p><p>Xn1</p><p>W:</p><p>Expi W:</p><p>Exp1 W:</p><p>Exp2 9</p><p>being:</p><p>W:</p><p>Exp1 Exp1 m: 4 RT4 kk 1 1 </p><p>p11p1</p><p> k1k</p><p> !10</p><p>W:</p><p>Exp2 Exp2 m: 12 RT12 kk 1 1 </p><p>p20p12</p><p> k1k</p><p> !11</p><p>In the analysis, fixed eciencies and no losses due to friction were assumed. Fixingthe liquid flow 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,</p><p>G. Cammarata et al. / Applied Energy 68 (2001) 1929 25</p></li><li><p>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 first turbine (respec-tively state 4, p4 p1, and 11, p11 p12):</p><p>FOM FOM p20;T 20;T 9; p4;T4; p12;T12 </p><p>12</p><p>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 nodes4 and 12, or on the thermo-pressure conditions of the input and output of thetwo expanders. In the configuration shown in the T-S diagram the FOM is equal to:</p><p>FOM </p><p>W: IDEALm:f</p><p> !W: REAL</p><p>m:f</p><p> ! T20 s20 s9 h20 h9W:</p><p>Comp P21</p><p>W:</p><p>Expi</p><p> 1m:f</p><p> 0:0593 5:93% 13</p><p>The liquid helium properties were obtained from McCarty [11].</p><p>4. Results</p><p>As described, the numerator of the FOM is constant, so to maximize the ratio it isenough to minimize the denominator. Particularly, since m</p><p>:f is a fixed value,</p><p>obtained from the operating conditions of the plant, it is possible to maximize FOMby minimizing the following objective function:</p><p>WR 1Comp</p><p>m: 20 RT20 ln p1p20</p><p> m: 4 k</p><p>k 1R</p><p> 1 p12p4</p><p> k1k</p><p>" #T4 Exp1 1 p20</p><p>p12</p><p> k1k</p><p>" #T12 Exp2</p><p>( )14</p><p>The range of values, in which it is possible to obtain the solution, has been deter-mined by the following considerations:</p><p>. T4 falls in the range 70100 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 first exchanger). The higher limit of the range depends on the low e-ciencies required for the two exchangers placed before section 4;</p><p>26 G. Cammarata et al. / Applied Energy 68 (2001) 1929</p></li><li><p>. p1 is acceptable if contained in the range 1.12.0 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;</p><p>. p12 is to be found in the range 0.30.7 MPa, where these values have been chosenin order not to eliminate the small enthalpic drop in one of the two turbines;</p><p>. T12 may vary over the range 1525 K, in order not to excessively penalise theeciency of the exchanger placed between sections 11 and 12.</p><p>Once defined, 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 fitness or to verify, to goodapproximation, that the value obtained is an absolute and not a local minimum inthe defined field 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, fixing 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 fitness (WR) and the corresponding characteristicvalues of the Genetic Algorithm are:</p><p>Table 1</p><p>Values of fitness, WR, varying the crossover index</p><p>Generations=200, Index tm 0:001Crossover index tc No. of members of population</p><p>10 30 50</p><p>0.6 100,007 99,9838 99,9769</p><p>0.7 100,007 100,007 99,9773</p><p>0.8 100,1315 99,9869 99,9777</p><p>Table 2</p><p>Values of fitness, WR, varying the mutation index</p><p>Generations=200, Index tc 0:6Mutation index tm No. of members of population</p><p>10 30 50</p><p>0.003 99,9876 99,994 99,996</p><p>0.009 100,019 99,978 99,994</p><p>0.018 99,988 99,9769 99,986</p><p>G. Cammarata et al. / Applied Energy 68 (2001) 1929 27</p></li><li><p>WR 99:9769 kW tc 0:6; tm 0:001 and ns 50</p><p>Once the values of the parameters corresponding to the minimum of the objectivefunction have been fixed, an additional verification consists in obtaining an asymp-totic law of the fitness trend versus the number of generations. As shown in Fig. 3,the value of fitness, WR, becomes constant after 200 generations, confirming, 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 previo...</p></li></ul>


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