GLOBAL DESIGN OPTIMIZATION OF A REFRIGERATION SYSTEM USING A GENETIC ALGORITHM. L. Govindarajan and T. Karunanithi

Transcription

1 GLOBAL DESIGN OPTIMIZATION OF A REFRIGERATION SYSTEM USING A GENETIC ALGORITHM L. Gvndarajan and T. Karunanth Department f Chemcal Engneerng Faculty f Engneerng and Technlgy Annamala Unversty, Inda cdl_lln@sancharnet.n ABSTRACT The ptmal desgn f ndustral three-stage refrgeratn systems t mnmze ttal prductn cst has been effectvely mplemented usng an genetc algrthm (GA), whch s an effcent alternatve fr cnventnal search algrthms. In ths artcle the glbal ptmum desgn parameters f a refrgeratn system btaned by usng GA s cmpared wth the Nelder-Mead smplex search algrthm. The results prve that glbal desgn ptmzatn usng GA s mre rbust and smple, as t requres n ntal guess values f desgn varables. Hence the prpsed technque s well suted fr desgnng a varety f ndustrally mprtant systems. OPSOMMING De ptmum ntwerp van 'n drestadum-vresssteem m ttale prduksekste te beperk, wrd deltreffend met behulp van 'n genetese algrtme (GA) as alternatef vr knvensnele sekalgrtmes n werkng gestel. De navrsng s daarp tegespts m de glbale ptmum ntwerpparameters van de GA met de Nelder-Mead Smpleks-sekalgrtme te vergelyk. De resultate tn dat de GA se glbale ptmum ntwerp rbuust en eenvudg s aangesen geskatte aanvangswaardes vr ntwerpveranderlkes ne bendg wrd ne. Derhalwe s de GA-tegnek besnder geskk vr de ntwerp van nywerhedssteme van uteenlpende aard. SA Jurnal f Industral Engneerng (2): 37-45

2 1. INTRODUCTION Refrgeratn fnds extensve applcatn n lubrcatng-l purfcatn, lw-temperature reactns, and separatn f vlatle hydrcarbns n petrleum ndustres. It s als used n the manufacture f ce and the dehydratn f gases n many prcess ndustres and n the ar cndtnng f buldngs. Refrgeratn plays a vtal rle n the treatment, transprtatn, and preservatn f fd, beverages, etc. Prcess refrgeratn s used at many dfferent temperature levels t cndense r cl gases, vaprs and lquds. The ptmal desgn f ndustral prcesses and equpment was carred ut wth the cnventnal mathematcal prgrammng technques that use algrthms devsed fr slvng general ptmzatn prblems (Knpf [1], Grssmann [2]). These algrthms are nly effectve fr mderately cnstraned ptmzatn prblems wth cntnuus decsn varables, whereas the ndustral prcess and equpment desgn prblems cntan dscrete, as well as cntnuus, varables and nvlve a large number f cnstrants. A heurstc methd was als reprted fr the desgn f ndustral equpment but ths methd may end n lcal ptmum values due t ts greedy nature. It s nt a general methd due t the fact that specal heurstc rules may be needed fr slvng the specfc ndustral desgn prblems (Md [3], Yeh [4]). Hence, these methds requre labrus and tme-cnsumng cmputatn steps t arrve at ptmum values n the desgn f ndustral prcesses and equpment. The ptmum desgn f the ndustral refrgeratn system s characterzed by the presence f nn-cnvex desgn spaces. The standard lnear, smplex search algrthm and descent technques were appled t the desgn f a refrgeratn system. The cmmnly used smplex search methd s the Nelder-Mead search algrthm, as t s sutable fr prblems that are very nnlnear and/r have a number f dscntnutes. Generally, these methds are neffcent, cmputatnally expensve, and n mst cases, fnd a lcal ptmum that s clsest t the ntal guess values. Recently, a glbal ptmzatn technque knwn as the genetc algrthm (GA) has becme a canddate fr many ptmzatn applcatns due t ts flexblty and effcency (Gldberg [5]). GA s a stchastc searchng algrthm that cmbnes artfcal survval f the fttest wth genetc peratrs abstracted frm nature t frm a surprsngly rbust mechansm that s sutable fr a varety f ptmzatn prblems. GA hps randmly frm pnt t pnt, thus allwng t t escape frm lcal ptmums n whch the ther algrthm mght land. Therefre, the GA-based glbal ptmum prblem can be apprached wth hgh prbablty. Ths artcle descrbes the desgn f a three-stage ndustral refrgeratn system usng the genetc algrthm and cnsderng the varus prcess cnstrants t mnmze the ttal cst nvlved n the peratn. 2. THE GENETIC ALGORITHM IN OPTIMAL DESIGN OF INDUSTRIAL EQUIPMENT The genetc algrthm s a cmputer-based search-and-ptmzatn technque based n the mechancs f natural genetcs and selectn. The ptmzatn prblem usng GA cnssts f the fllwng steps: 38

3 1. Cdng and decdng f prblem varables. 2. Frmulatn f the ftness functn. 3. Genetc peratns. The genetc algrthm fnds extensve applcatn n a varety f prblems n ndustry. GA s used n large-scale ndustral energy ntegratn (Yu [6]), ptmal desgn f multprduct batch chemcal prcesses (Wang [7], Har [8]), ptmzatn f chemcal flw shp sequencng (Cartwrght [9]), sequencng f batch peratns fr a hghly cupled prductn prcess (Lhl [10]), ptmzatn f peratns n an ndustral cgeneratn system (Manlas [11]), and desgn batch semcntnuus prcess plant treatng multple prducts. 3. PROBLEM FORMULATION Optmzatn f Industral Refrgeratn System The refrgeratn system shwn n Fgure 1 has three stages n whch a ht stream s cled by usng lqud refrgerant. Each stage cnssts f a heat exchanger wth ht stream n ne sde f the heat exchange surface and a blng refrgerant n the ther sde. The temperatures at whch the refrgerants bl are knwn at each stage and hence the rate f heat transfer s determned by the area f the heat exchange surface, at a gven flw rate and nlet temperature f the ht flud. The ptmal desgn prblem necesstates the determnatn f the surface area f three heat exchangers requred t cl a flud wth a specfc heat f J/kg C flwng at a rate f 10,800 kg per hur frm 10 C t -55 C. The unt s expected t perate fr a mnmum f 300 days n a year. The system parameters used n the desgn are the latent heat f the refrgerant (λ), 2,32,600 J/Kg, and the verall heat transfer ceffcent n stages, 1130 J / s m 2 C. 39

4 The desgn bjectve s t mnmze the ttal cst fr the three stages n the refrgeratn system gven by C = [ a ( A ) + bw ] (1) = 1 The frst term represents the captal cst related t the area f heat exchange surface and the secnd term s the peratng cst f supplyng the refrgerant. The purpse f ptmzatn s t btan reasnable values f desgn varables.e. temperatures f flud leavng the stages 1 and 2. The area f heat exchange and the lqud refrgerant addtn rate n each stage are determned by usng the desgn varables. The lqud refrgerant temperature n each stage s T 1 = 18 C; T 2 = 40 C ; T 3 = 62 C ; The temperature at whch the flud enters the system (T 0 ) s 10 C and the temperature at whch t leaves the system (T 3 ) s -55 C. The temperature leavng the gven stage must be greater than the refrgerant temperature. Therefre, the cnstrants n the desgn varables are T0 = 10 C T1 18 C (2) T 40 C (3) T1 2 The heat transfer rate at stage : Q = U A T (4) ( ) ln The lg mean temperature at stage : ( ) ( T 1 T ) T ln = (5) ( T 1 TR ) ln ( T TR ) The energy balance ver refrgerant ndcates Q = λ W (6) The energy balance ver the flud ndcates Q = F Cp T T (7) GA Slutn Methdlgy Cdng ( ) 1 In the genetc algrthm, the desgn varables are frst cded n strng structures. Bnary cded strngs havng 0 and 1 are wdely used. The desred slutn accuracy determnes the length f the strng. A pnt n the search space s represented by an eght-bt strng and s allcated accrdng t a lnear mappng rule: 40

5 max mn X j X mn j j = X j + D L k (8) j X 2 1 The peratn f GA begns wth a ppulatn f randm strngs representng the desgn varable. Each strng s evaluated t fnd the ftness functn. The three man GA peratrs - reprductn, crssver and mutatn, are appled t the randm ppulatn t create a new ppulatn. The ppulatn s evaluated and tested untl the termnatn crtern s met, teratvely altered by the GA peratrs. Generatn n GA represents the cycle f peratn by the genetc peratrs and the evaluatn f the ftness functn. Reprductn Reprductn selects gd strngs n a ppulatn and frms a matng pl. The reprductn peratr wll pck the abve-average strngs frm the current ppulatn. Ther cpes are then nserted nt the matng pl n a prbablstc manner. The strng wth a hgher ftness value wll represent a larger range n the cumulatve prbablty values and therefre has a hgher prbablty f beng cped nt the matng pl. On the ther hand, a strng wth a smaller ftness value represents a smaller range n the cumulatve prbablty values and has a smaller prbablty f beng cped nt the matng pl. Crssver In the crssver peratn, exchangng nfrmatn between strngs f the matng pl creates new strngs. The tw strngs partcpatng n the crssver peratn are knwn as parent strngs and the resultng strngs are knwn as chld strngs. The chld strngs prduced may be gd r nt, whch depends n the perfrmance f the crssver ste. The effect f crssver may be benefcal r detrmental. In rder t preserve sme gd strngs that are already present n the matng pl, nt all strngs n the matng pl are used n the crssver. Mutatn Mutatn s needed t create a pnt n the neghbrhd f the current pnt, thereby achevng lcal search arund the current slutn. The mutatn s als used t mantan dversty n the ppulatn. Frmulatn f the Ftness Functn The bjectve s t mnmze the ttal cst, subject t varus cnstrants gvernng the prcess. A penalty functn apprach s used t handle the explct cnstrants. Penalty terms are ncrprated n the ftness functn and are sad t reduce the ftness f the strng accrdng t the magntude f ther vlatn N Mnψ = [ a ( A ) + bw ] + λ C z Cz (9) (lmt) = 1 z LVC The ftness s btaned by transfrmng the mnmzatn prblem nt a maxmzatn prblem as 41

6 1 FIT = (10) 1 + ψ 4. RESULTS AND DISCUSSION The prpsed GA-based ptmzatn has been successfully ncrprated fr the ptmal desgn f a three-stage refrgeratn system. The cst ceffcents f three-stage the refrgeratn system are gven n Table 1. Cst Ceffcents Stages Captal cst parameters (a), cst unts / s(m 2 ) ½ Operatng cst parameters (b), cst unts / Kg Table 1. Cst Ceffcents fr an Industral Three-Stage Refrgeratn System The desgn prblem has been mplemented usng the Nelder-Mead Smplex search algrthm wth the MATLAB ptmzatn tlbx. The varatn f the ttal cst fr varus ntal guesses f desgn varables has been studed and the results are gven n Table 2. Intal Guess f Desgn Parameters Optmzed Desgn Values Ttal Cst T 1 C T 2 C T 1 C T 2 C Cst Unts/Year Less than -1 Less than 3 Slutns dverge Abve -9.5 Abve 37 Slutns dverge Table 2: Influence f Intal Guess Values n Optmal Desgn Parameter usng the Nelder-Mead Smplex Search Methd It can be cncluded that ths type f mplementatn requres a gd startng pnt. Hwever, n cntnuus prcess peratns, t s nt pssble t predct a sutable startng pnt t btan the ptmum desgn. Cnventnal search methds such as the Nelder-Mead smplex search algrthm arrve at the lcal ptmum values nearer t the ntal guess f desgn varables. 42

7 The prpsed methd f GA-based ptmal desgn vercmes the shrtcmngs f the cnventnal search methd as t starts wth a ppulatn representng many pnts. The results f the prpsed methd have been cmpared wth the Nelder-Mead Smplex search algrthm and are gven n Table 3. Optmzatn Methd Desgn Parameters A 1 A 2 A 3 W 1 m 2 m 2 m 2 Kg/hr W 2 Kg/hr W 3 Kg/hr Ttal Cst Cst Unts/Year Nelder-Mead Smplex Search Genetc Algrthm ,02, ,02,243 Table 3: Cmparsn between the Nelder-Mead Smplex Search Methd and Genetc Algrthm-based Optmzatn Technque The results ndcate that the desgn values btaned frm GA shw mprvement n the mnmzatn f the ttal cst ver thse btaned by the cnsdered cnventnal smplex algrthm and the same results are expected t be btaned by applyng any ther cnventnal mathematcal technques. The prpsed GA-based desgn ptmzatn can be successfully mplemented fr the desgn f ndustral three-stage refrgeratn prcess systems, due t the fllwng advantages: 1. GA has the ablty t gve slutns near enugh t glbal ptmal values. 2. GA s effcent n handlng prblems havng dscrete varables and nn-cnvex behavur, whch are cmmn characterstcs encuntered n the desgn f ndustral equpments. 3. Intal guesses f the ptmzatn varables are nt requred fr GA, whereas the Nelder-Mead-lke smplex algrthm ntal estmate and descent drectn are very mprtant fr btanng feasble slutns. 4. GA, due t ts ntellgent apprach, results n faster cnvergence when cmpared t the cnventnal mathematcal, heurstcs and ther artfcal ntellgence technques such as smulated annealng. 5. GA s smple n structure and cnvenent fr mplementatn n the desgn f largescale ndustral equpment. 5. CONCLUSION The ptmal desgn f ndustral equpment such as a three-stage refrgeratn system has been successfully mplemented usng the genetc algrthm. The genetc algrthm ffers sgnfcant advantages ver the cnventnal mathematcal prgrammng methds and heurstc ptmzatn technques as t prvdes glbal desgn parameters and s effectve n handlng prblems wth dscrete varables havng a large number f cnstrants. The genetc algrthm can be used as a unversal technque fr slvng a varety f ndustral desgn prblems. 43

8 Ntatn a Captal cst ceffcent, cst unts/s (m 2 ) 0.5 A Area f heat exchange surface, m 2 b Operatng cst ceffcent, cst unts/kg Cp Specfc heat f flud, J/kg C C Ttal cst, cst unts/year D Equvalent decmal value F Ht flud flw rate, Kg/hr FIT Ftness functn GA Genetc algrthm LVC Lmt vlated cnstrant Q Rate f heat transfer, J/s T Temperature, C U Overall heat transfer ceffcent, J/s m 2 C W Lqud refrgerant addtn rate, Kg/hr X Desgn varable Subscrpts j k R Stages Desgn varable Sub-strng Refrgerant Superscrpts L lmt mn, max z Number f bnary bts n a sub-strng Cnstrant lmt Lmts f the desgn varables Cnstrant functn Prta Matseke {WITS} Specal Symbls λ λ ψ Penalty factr Latent heat f refrgeratn, J/Kg Objectve functn t be mnmzed 6. REFERENCES [1] Knpf F.C, Oks M.R & Reklats G.V, (1982), Optmal Desgn f Batch/Sem- Cntnuus Prcesses, Industral Engneerng Chemstry Prcess Desgn and Develpment, Vl 21, p [2] Grssmann I., & Sargent R.W.H, (1979), Optmum Desgn f Multpurpse Chemcal Plants, Industral Engneerng Chemstry Prcess Desgn and Develpment, Vl 18, p , [3] Md A.K &Karm A, (1989), Desgn f multprduct batch prcesses wth fnte ntermedate strage, Cmputers and Chemcal Engneerng, Vl 13, p

9 [4] Yeh N.C & Reklats. G.V, (1987) Synthess and szng f batch/ sem cntnuus prcess: Sngle prduct plants, Cmputers and Chemcal Engneerng, Vl 11, p [5] Gldberg D.E., Genetc Algrthms n Search, Optmzatn and Machne Learnng, Addsn - Wesley, New Yrk. [6] Yu H, Fang H, Ya P & Yuan Y, A cmbned genetc algrthm/smulated annealng algrthm fr large scale system energy ntegratn, Cmputers and Chemcal Engneerng, Vl 24, p , [7] Wang C, Quan H & Xu X, Optmal desgn f mult prduct batch chemcal prcess usng genetc algrthms, Industral Engneerng Chemstry Research, Vl 35, p ,1996. [8] Har L.B, Pantel C.A, Dmenech S & Pbuleau L, Desgn f multpurpse batch chemcal plants usng a genetc algrthm, Cmputers and Chemcal Engneerng, Vl 22, S777-S780, [9] Cartwrght H.M & Lng R.A, Smultaneus ptmzatn f chemcal flw shp sequencng and tplgy usng genetc algrthm, Industral Engneerng Chemstry Research, 32, ,1993. [10] Lhl T, Sculz, C Engell S, Sequencng f batch peratns fr a hghly cupled prductn prcess: genetc algrthms versus mathematcal prgrammng, Cmputers and Chemcal Engneerng, 22, S579-S585, [11] Manlas D.A, Galamass T.P, Frangpuls C.A & Tsahals D.T, A genetc algrthm fr peratn ptmzatn f an ndustral cgeneratn system, Cmputers and Chemcal Engneerng, 20, S1107-S1112,

10 46