Unconstrained Gibbs free energy minimization for phase equilibrium calculations in non-reactive systems using an improved Cuckoo Search algorithm

Transcription

1 Insuo Tecnologco de Aguascalenes From he SelecedWorks of Adran Bonlla-Percole 2014 Unconsraned Gbbs free energy mnmzaon for phase equlbrum calculaons n non-reacve sysems usng an mproved Cuckoo Search algorhm Sef E.K. Faeen, Caro Unversy Adran Bonlla-Percole Avalable a: hps://works.bepress.com/adran_bonlla_percole/305/

2 pubs.acs.org/iecr Unconsraned Gbbs Free Energy Mnmzaon for Phase Equlbrum Calculaons n Nonreacve Sysems, Usng an Improved Cuckoo Search Algorhm Sef-Eddeen K. Faeen Deparmen of Chemcal Engneerng, Caro Unversy, Gza, Egyp Adrań Bonlla-Percole* Deparmen of Chemcal Engneerng, Insuo Tecnoloǵco de Aguascalenes, Aguascalenes, Me xco ABSTRACT: Ths sudy nroduces a sraegy o mprove he effecveness of Cuckoo Search (CS) algorhm for he unconsraned Gbbs free energy mnmzaon n phase equlbrum calculaons of nonreacve sysems. Specfcally, he graden nformaon of he unconsraned Gbbs free energy funcon, whch s readly avalable, s used o enhance he balance beween dversfcaon and nensfcaon sages of he CS algorhm for phase-spl calculaons n mulcomponen sysems. The resuls showed ha s feasble o mprove he numercal performance of he CS algorhm usng he graden nformaon of he Gbbs free energy funcon; hs mproved mehod provdes beer resuls for phase equlbrum calculaons n nonreacve sysems wh nsgnfcan addonal compuaonal effor. Ths graden-based Cuckoo Search (GBCS) algorhm ouperformed he convenonal CS algorhm, n erms of s relably and effcency n solvng phase equlbrum problems, especally for mulcomponen sysems. 1. INTRODUCTION The mnmzaon of he Gbbs free energy for he predcon of phase behavor of a mulcomponen mxure s one of he mos mporan and challengng global opmzaon problems, n he conex of appled hermodynamcs for chemcal engneerng. Ths opmzaon problem can be solved usng boh consraned and unconsraned formulaons, 1,2 where he objecve s o deermne he number, ype, and composon of he phases a equlbrum under he gven operang condons. Opmzaon mehods for mnmzaon of he Gbbs free energy n nonreacve sysems employ local or global solvng sraeges. 1,2 They nclude he acceleraed successve subsuon mehod, Nelson s mehod, he lnearly consraned mnmzaon mehod, homoopy connuaon algorhms, and deermnsc and sochasc global opmzaon sraeges. 1 7 Overall, several opmzaon sraeges do no provde a guaranee of obanng he global mnmum of Gbbs free energy funcon, because of s mulvarable, nonlnear, and nonconvex naure. 1 3 Consderng hese challengng characerscs, s easy o undersand he dffculy of fndng a global mnmum of he Gbbs free energy n phase equlbrum calculaons (PEC). In parcular, phase-equlbrum problems have been successfully solved usng opmzaon meaheurscs. 2,8 17 These mehods combne heurscs n hgh-level frameworks o explore effecvely he search space for fndng he global opmum soluon. They offer several advanages for performng he global mnmzaon of Gbbs free energy, because hey are no problem-specfc, may use a long-/shor-me form of memory o gude he search for he global opmzaon avodng he convergence o local opma, and may use doman-specfc knowledge of he problem o explore he bes promsng areas. 2 These opmzaon sraeges have been esed and evaluaed on several phase equlbrum problems wh vapor lqud equlbrum (VLE), lqud lqud equlbrum (LLE), and vapor lqud lqud equlbrum (VLLE) Meaheurscs used for PEC nclude he mos mporan and popular sngle-pon and populaon-based mehods: Smulaed Annealng, Genec Algorhms, Tabu Search, Dfferenal Evoluon, Parcle Swarm Opmzaon, Frefly Algorhm, An Colony Opmzaon, Cuckoo Search, and oher algorhms In parcular, Bhargava e al. 16 found ou ha Cuckoo Search (CS) s one of he mos relable meaheurscs for he unconsraned Gbbs free energy mnmzaon n PEC of nonreacve sysems. Heren, s mporan o remark ha he numercal performance and convergence behavor of opmzaon meaheurscs are deermned by boh he nensfcaon and dversfcaon sraeges, whch gude and modfy he numercal operaons o effcenly explore he search space of decson varables n order o fnd (near-)global opmal soluons. Specfcally, he dversfcaon (exploraon) sage s devoed o explorng he search space, and for he compuaon of he objecve funcon value of very dfferen pons n he search doman o fnd he promsng area. On he oher hand, he nensfcaon (exploaon) sage nvesgaes he promsng area o locae, as near as possble, he global opmum soluon ha he algorhm s lookng for. Alhough n recen years, here have been sgnfcan advanages n he applcaon of meaheurscs o perform he Gbbs free energy mnmzaon for PEC, fndng a good balance beween dversfcaon and Receved: Aprl 22, 2014 Revsed: June 15, 2014 Acceped: June 17, 2014 Publshed: June 17, Amercan Chemcal Socey 10826

3 nensfcaon sages of curren meaheurscs s sll a challenge, especally for mulcomponen sysems. The lack of he proper balance of hese sages may lead o a poor convergence performance and low algorhm relably. Ths problem prevals even for he mos promsng meaheurscs such as Cuckoo Search or Bare-Bones Parcle Swarm Opmzaon, whch may show some numercal dsadvanages for PEC. 13,16 The desrable scenaro for he applcaon and mplemenaon of a meaheurscs for he global mnmzaon of he Gbbs free energy funcon n a process smulaor s ha he opmzaon mehod can quckly denfy regons n he search space wh hgh-qualy soluons, whou wasng oo much compuaonal me n unpromsng regons. In parcular, a recen sudy 18 showed ha CS s a relable sochasc opmzaon mehod for solvng he unconsraned Gbbs mnmzaon problem and may offer a beer performance han oher sae-of-he-ar sochasc mehods. However, also showed ha here s a need o mprove he effcency of he CS algorhm, especally for mulcomponen sysems. Based on hese facs, hs sudy s focused on he developmen of a sraegy o mprove he effecveness of he CS algorhm for he unconsraned Gbbs free energy mnmzaon n PEC of nonreacve sysems. Ths mprovemen sraegy s based on he use of he graden nformaon of he unconsraned Gbbs free energy funcon o enhance he balance beween dversfcaon and nensfcaon sages of he CS algorhm for PEC. One of he mos commonly used and wellsuded conceps n numercal opmzaon s he graden. The graden a any pon n he decson space ndcaes he drecon n he search space along whch he funcon o be opmzed mproves he mos. 19 Therefore, hs drecon, whch s derved from he objecve funcon graden, can be ncorporaed no he meaheurscs o mprove her numercal performance. Our resuls showed ha s feasble o mprove he performance of he CS algorhm usng he graden nformaon of he Gbbs free energy funcon. Ths mproved mehod provdes beer resuls for PEC n nonreacve sysems whou addng a compuaonal complexy o he opmzaon algorhm. In summary, hs sudy nroduces a modfcaon o he CS algorhm, whch manans he relably of he algorhm bu sgnfcanly enhances s effcency n he unconsraned Gbbs free energy mnmzaon for PEC n mulcomponen sysems. 2. THE UNCONSTRAINED GIBBS FREE ENERGY MINIMIZATION FOR PHASE EQUILIBRIUM CALCULATIONS IN NONREACTIVE SYSTEMS In a phase equlbrum problem, a mxure of subsances a a gven emperaure (T), pressure (P), and oal molar amoun may separae no wo or more phases. The global mnmzaon of he Gbbs free energy mus be performed for calculang he equlbrum sae of a mxure. 1,2 Ths approach was nroduced by Whe e al. 20 and, o dae, many opmzaon sraeges have been suded for he global mnmzaon of hs hermodynamc funcon. 1 3 For a sable equlbrum a gven T and P, he Gbbs free energy of mxng s he hermodynamc objecve funcon, whch mus be a he global mnmum. Ths opmzaon problem can be defned as Δ g g g mn = RT RT subjec o 0 = π c Δ μ j nj RT 1 1 (1) j= = π nj = zn F = 1,..., c j= 1 (2) 0 n zn = 1,..., c; j = 1,..., π j F (3) where n j and Δμ j /RT are he mole numbers and chemcal poenals of speces n phase j, respecvely; c and π are he numbers of componens and phases a equlbrum, respecvely; z s he mole fracon of componen n he feed; and n F s he oal moles n he feed. Decson varables n j for hs consraned opmzaon problem are c π. The unconsraned mnmzaon of he Gbbs free energy funcon can be performed usng alernave varables χ j nsead of n j as decson varables. 8 For mulphase nonreacve sysems, new decson varables χ j (0,1) can be defned and employed by usng he followng expressons for he mass balances: n1 = χ zn = 1,..., c 1 F (4) j 1 n = χ ( zn n ) = 1,..., c; j = 2,..., π 1 j j F m= 1 n = zn n = 1,..., c π F π 1 m= 1 m m where he number of decson varables χ j s c(π 1) for he unconsraned Gbbs free energy mnmzaon of nonreacve sysems. In mos of he repored sudes, auhors assumed ha he number and ype of phases a equlbrum are known; such problems are also refereed as phase spl calculaons. In hs sudy, he same assumpon s made. For example, for a wophase spl problem, he objecve funcon becomes Δ c = Δμ c g Δ + μ α β nα nβ RT RT = = RT 1 1 (7) where he subscrps α and β refer o he wo phases a equlbrum. The decson varables χ α for a wo-phase spl problem are relaed o he mass balances as follows: n = χ zn = 1,..., c α α F (8) n = zn χ zn = 1,..., c On he oher hand, he graden of he unconsraned Gbbs free energy funcon can be easly derved as follows: Δ π c 1 g Δμ j nj = RT χ χ α = = RT k T j 1 1 kα π c Δ ( μ / RT) j + nj k = 1,..., c j= 1 = 1 χk α (10) β F α F (9) The dervave of he number of moles n α and n β, wh respec o he decson varables χ α,s n = α zn k F = χ 0 k (11) kα nβ = zn F = k χ 0 k kα (5) (6) (12) 10827

4 Fgure 1. Pseudocode of he graden-based cuckoo search (GBCS) algorhm used for he unconsraned mnmzaon of he Gbbs free energy n nonreacve sysems. The second erm of eq 10 s dencal o zero, based on he Gbbs Duhem equaon. Thus, he graden of he unconsraned objecve funcon of a wo-phase spl problem (e.g., eq 7), s analycally obaned as 1 Δg RT χ kα Δμk = zn k F RT α μ Δ kβ k = 1,..., c RT (13) I s mporan o remark ha he evaluaon of eq 13 (.e., he graden of Δg/RT) does no nvolve any dervaves of he chemcal poenal funcon and can be easly mplemened ndependen of he hermodynamc model used (.e., EoS or local composon model). Fnally, he compuaonal cos assocaed wh he calculaon of hs graden s nsgnfcan, from a praccal pon of vew, snce he chemcal poenals are calculaed n he course of he calculaon of he objecve funcon. We wll llusrae ha he use of he graden of he Gbbs free energy funcon may ncrease he effecveness of meaheurscs for PEC, because provdes he drecon of maxmum mprovemen o search for he opmum of he objecve funcon. Based on hs fac, n he followng secon, we descrbe he approach used o mprove he numercal performance of Cuckoo Search for he unconsraned mnmzaon of Gbbs free energy n wo-phase equlbrum calculaons of nonreacve sysems. 3. STRATEGY TO IMPROVE THE BALANCE OF DIVERSIFICATION AND INTENSIFICATION STAGES OF CUCKOO SEARCH FOR THE UNCONSTRAINED MINIMIZATION OF THE GIBBS FREE ENERGY In hs sudy, we have used he Cuckoo Search (CS) as an opmzaon algorhm for he unconsraned mnmzaon of he Gbbs free energy for PEC n nonreacve sysems. The use of CS and s varans for solvng mulmodal opmzaon problems s ganng wde populary, because of s ease of use and effecve performance Ths meaheursc emulaes he brood parassm behavor of he cuckoo brds, whch lay her eggs n he ness of oher speces. Ths naure parassm behavor mples ha he cuckoo eggs evolved o mmc he egg appearance of local hos brds o avod beng dscovered and abandoned. Therefore, he numercal mplemenaon of CS for global opmzaon s based on he followng rules: (1) Each cuckoo lays an egg n a random nes. The egg represens a se of soluons for he opmzaon problem. (2) The bes eggs (.e., soluons) are conaned n a fracon of he ness and are carred over o he nex eraon. (3) The number of ness does no change. A hos brd can fnd an alen egg wh a specfed probably p a [0,1]. If an alen egg s found, he hos can abandon he nes or dscard he egg, and hen buld a new nes elsewhere. Ths condon s mplemened n CS wh he assumpon ha a fracon p a of n h ness s replaced by new ness. The use of Le vy flghs n he CS algorhm makes s local and global search sages effecve. A Le vy flgh s a rajecory ha consss of akng successve random seps. Ths sequence of sudden jumps, whch are chosen from a power-law al probably densy funcon, s a characersc of he Le vy flgh, whch s consdered as he opmum random search paern. To generae a new egg, a Le vy flgh s performed usng he coordnaes of a randomly seleced egg. Ths sep can be represened by + X 1 = X + α Levy( λ) (14) where denoes enry-wse mulplcaon, α s he sep sze, and Le vy (λ) s he Le vy dsrbuon. The condon for dsplacng an egg o he new poson s ha he value of he objecve funcon s found beer han anoher randomly seleced egg. The sep sze α, whch depends on he scale of he opmzaon problem, conrols he scale of random search. A fracon (1 p a ) of he ness seleced a random s abandoned and replaced by new ones a new locaons va local random walks: + 1 j k X = X + α( X X ) (15) where X j and X k are wo dfferen soluons seleced randomly by random permuaon and α s a random number drawn from a unform dsrbuon. The only parameer o be uned s he fracon of ness o be abandoned (1 p a ). However, hs value s no crcal for he opmzaon algorhm and Yang and Deb 21 suggesed usng a value of p a =

5 Table 1. Descrpon, Thermodynamc Models, and Feed Condons of Seleced Phase Equlbrum Problems No. sysem feed condons hermodynamc models global opmum 1 n-buyl aceae + waer n F = (0.5, 0.5) a 298 K and kpa NRTL model oluene + waer + anlne n F = ( , , ) a 298 K and kpa NRTL model N 2 +C 1 +C 2 n F = (0.3, 0.1, 0.6) a 270 K and 7600 kpa SRK EoS wh classcal C 1 +H 2 S n F = (0.9813, ) a 190 K and 4053 kpa SRK EoS wh classcal C 2 +C 3 +C 4 +C 5 +C 6 n F = (0.401, 0.293, 0.199, , ) a 390 K and 5583 kpa SRK EoS wh classcal C 1 +C 2 +C 3 +C 4 +C 5 +C 6 +C 7 16 n F = (0.7212, , , , , , , ) a SRK EoS wh classcal C K and kpa 7 C 1 +C 2 +C 3 +C 4 +C 4 +C 5 +C 5 + n F = (0.614, , , , , , , , SRK EoS wh classcal C 6 +C ) a 314 K and kpa 8 C 1 +C 2 +C 3 +C 4 +C 5 +C 6 +C 7 + C 8 +C 9 +C 10 n F = (0.6436, , , , , , , , , ) a K and kpa SRK EoS wh classcal Our prevous sudy 24 has ndcaed ha he convergence performance of CS can be mproved va he modfcaon of he local random walk n whch a fracon (1 p a ) of he ness s replaced. In parcular, he graden of he objecve funcon can be employed o deermne he sep drecon and magnude for generang new soluons. 24 Usng he graden nformaon, new ness are generaed randomly from he worse ness bu n he drecon of he mnmum as seen from he pon of vew of he old ness. Therefore, he new soluons can be obaned usng sep = α( X X ) j k sep + 1 X = X + sep sgn df (16) (17) where he sgn funcon obans he sgn of s argumen and df s he graden of he objecve funcon a each decson varable, f/ x. Noe ha f he objecve funcon graden s negave, he sep drecon s made posve; oherwse, he sep drecon s made negave. I s mporan o remark ha radonal meaheurscs do no ulze he graden nformaon n her dversfcaon and nensfcaon schemes. On he oher hand, eqs 16 and 17 do no aler he srucure of he opmzaon algorhm and no addonal parameers are needed o mplemen hem. In a prevous sudy, 16 CS algorhm has proven s superory over oher algorhms, n erms of relably (.e., he ably o fnd he global mnmum), for PEC. However, s degree of dversfcaon and nensfcaon s far from opmal, especally n challengng PEC. Noe ha, n he vcny of he global mnmum of a phase equlbrum problem, he orgnal CS s advanageous over oher mehods, because of s use of he local random search. 18 Prelmnary calculaons performed usng CS wh eqs 16 and 17, nsead of eq 15 n dfferen phase equlbrum problems suggesed ha s more convenen o use an adapve scheme for he generaon of new soluons, whch may combne he graden nformaon and he local random walk. Thus, we propose ha eqs 16 and 17 are used f he graden of he unconsraned Gbbs free energy funcon was >10 3 ; oherwse, he radonal local random search s appled n he CS algorhm. Pseudocode for hs mproved algorhm (graden-based cuckoo search, GBCS) s gven n Fgure 1, whch has been coded n MATLAB. In he followng secon, we wll compare he convergence performance of boh CS and GBCS n seleced PEC. Heren, s convenen o remark ha a comparson beween a se of promsng sochasc mehods for solvng phase equlbrum problems has been recenly publshed. 18 Is oucome clearly showed ha CS s a relable mehod for solvng phase sably and phase equlbrum problems and suggesed furher research o be dreced oward he mprovemen of he effcency of CS, especally for mulcomponen sysems. Thus, n hs sudy, we lmed he comparson of he resuls beween he orgnal CS algorhm and he developed modfed verson of CS, whch made use of he readly avalable graden of he unconsraned Gbbs free energy funcon, o probe he mprovemens n algorhm effcency. 4. RESULTS AND DISCUSSION In hs sudy, he wo opmzaon algorhms (CS and GBSC) and he hermodynamc models were coded n MATLAB. For he comparson of wo mehods, we have used p a = 0.25 and n h = 10n var, where n var s he number of decson varables. We suded egh PEC problems, whose deals can be found n Table 1. These problems are mulmodal wh he number of decson varables rangng from 2 o 10. Noe ha hese phase equlbrum problems have been used n prevous sudes for esng oher global opmzaon sraeges, ncludng classcal meaheurscs. 8,10,12,13,15,16 Each problem was solved 100 mes ndependenly wh a dfferen random number seed, for a robus performance analyss. GBCS and CS were evaluaed accordng o he relably and effcency for fndng he global opmum of he unconsraned Gbbs free energy funcon. The effcency s deermned by recordng he number of funcon evaluaons (NFE) for each opmzaon algorhm, where a low value of NFE means a hgher effcency. Noe ha NFE s an unbased ndcaor of he compuaonal coss requred by a ceran algorhm and s ndependen of he hos hardware. On he oher hand, he relably was measured by he success rae (SR) a ceran numercal effor. The success rae s defned as he rao of number of runs n whch he global mnmum was aaned whn a olerance a hs numercal effor o he oal number of runs. In addon, we also repor a plo of he average bes value agans NFE. The bes values are averaged over all he runs and ploed agans NFE, whch was calculaed a each eraon. Snce he NFE needed for each eraon dffer among he meaheurscs, he plo of average bes value agans NFE s a useful ndcaon of relably versus effcency of he opmzaon mehod. Fnally, he performance profles 26 for he relably and effcency 10829

6 Fgure 2. Evoluon of he mean bes funcon value wh NFE for CS and GBCS n he unconsraned Gbbs free energy mnmzaon of nonreacve sysems mercs of boh CS and GBCS have been calculaed usng he followng equaons: r psm = psm mn{ psm : 1 sm n s } (18) 1 ρ() ς = p R r ς s sze{ : psm } n p (19) where n s s he number of meaheurscs (.e., 2) o be esed, n p s he number of problems used n hs comparson, psm s he value 10830

7 Table 2. Mnmum NFE for he Average Bes Value To Reach a Tolerance Value (ε) from he Known Global Mnmum Usng CS and GBCS n he Unconsraned Mnmzaon of Gbbs Free Energy Mnmum Number of Funcon Evaluaons, NFE a olerance, ε PEC No. 1 PEC No. 2 PEC No. 3 PEC No. 4 PEC No. 5 PEC No. 6 PEC No. 7 PEC No. 8 CS Mehod GBCS Mehod a Boldface numbers represen he more effcen algorhm. Table 3. Success Rae of CS and GBCS n he Unconsraned Mnmzaon of he Gbbs Free Energy a Dfferen Ieraons Success Rae (%) a eraon PEC No. 1 PEC No. 2 PEC No. 3 PEC No. 4 PEC No. 5 PEC No. 6 PEC No. 7 PEC No. 8 CS Mehod GBCS Mehod a Boldface numbers represen he more effcen algorhm. of he performance merc for problem p and meaheursc sm, r psm s he performance rao used o compare he performance on problem p by meaheursc sm wh he bes performance by any meaheursc on hs problem; and ρ s (ς) s he fracon of he oal number of problems for whch meaheursc sm has a performance rao r psm whn a facor of ς of he bes possble rao. Noe ha r psm = 1 for he meaheursc ha performs he bes on a specfc problem p. The relably of he meaheursc n accuraely fndng he global mnmum of he unconsraned Gbbs free energy funcon s consdered as he prncpal goal; hence, he merc used o oban he relably performance profle s defned as psm = Δ g Δ* g RT RT calc (20) where f* and f calc s he known global mnmum and he mean value of he unconsraned Gbbs free energy funcon (.e., f = Δg/RT) calculaed by he meaheursc over several runs. Ths relably performance profle compares how accuraely CS and GBCS can fnd he global opmum value of Δg/RT, relave o each oher. We have also used anoher performance merc for he evaluaon of he effcency of boh CS and GBCS n obanng he global mnmum of he unconsraned Gbbs free energy. Ths merc s he mnmum number of NFE needed o reach whn 10 5 of he global mnmum of Δg/RT. Ths effcency performance profle compares how fas CS and GBCS can fnd he global mnmum wh a olerance level of 10 5 and s useful o denfy he meaheursc ha reaches he soluon fases for he phase equlbrum problems esed. The performance resuls of boh CS and GBCS are presened n hree dfferen ways. For each problem, he mean bes values are ploed versus NFE for each of he egh phase equlbrum problems. These plos are repored n Fgure 2. The mnmum NFE requred o reach a ceran olerance from he known global mnmum for each problem were calculaed and presened n Table 2, whle he relably (.e., success raes) of boh meaheurscs for solvng PEC are repored n Table 3 and Fgure 3, whch shows he success rae a dfferen number of eraons. For he purpose of hs sudy, he success rae s defned as he percenage of runs ha converged o whn 10 6 of he known global mnmum. Fnally, he performance profles of CS and GBCS for he relably and effcency mercs are 10831

8 Fgure 3. Global success raes (GSR, %) of he CS and GBCS algorhms n he unconsraned mnmzaon of he Gbbs free energy for he phase equlbrum calculaons (PEC) of nonreacve sysems. shown n Fgures 4 and 5, respecvely. A bref dscusson of he resuls follows. Fgure 4. Relably performance profles of he CS and GBCS algorhms n he unconsraned mnmzaon of he Gbbs free energy for he phase equlbrum calculaons (PEC) of nonreacve sysems. PEC No. 1 s a wo-componen lqud lqud equlbrum (LLE) problem ha s relavely easy o solve. The wo algorhms were able o solve he problem sasfacorly o he level of 10 7 dsance from he global mnmum of Δg/RT whn 2000 NFE, as ndcaed n Table 2. GBCS was slghly beer han CS n performng PEC, achevng a 100% success rae a 50 eraons, bu he savngs n compuaonal me s no sgnfcan, as depced n Fgure 2a. PEC No. 2 s also an LLE problem for a ernary sysem and he convergence paern o he global mnmum of he wo algorhms s smlar o ha for PEC No. 1. Boh algorhms performed he unconsraned mnmzaon of he Gbbs free energy sasfacory wh no sgnfcan savngs n compuaonal me for GBCS, as ndcaed by he mnmum NFE requred o reach a ceran dsance from he global mnmum as shown n Table 2. GBCS and CS reached a 100% success rae a eraons 300 (see Table 3). GBCS converged o he global mnmum of Δg/RT for PEC No. 3, whch s a ernary sysem wh vapor lqud equlbrum (VLE), n a sgnfcanly less NFE, as shown n Table 2 and Fgure Fgure 5. Effcency performance profles of he CS and GBCS algorhms n he unconsraned mnmzaon of he Gbbs free energy for he phase equlbrum calculaons (PEC) of nonreacve sysems. 2c. The savngs n compuaonal effor ranges from 60% a he 10 3 level o 13% a he 10 7 level, wh respec o he resuls obaned wh CS. Noe ha GBCS showed 100% relably a 100 eraons. PEC No. 4 s a bnary sysem wh VLE and he performance resuls ndcaed ha GBCS dd no ouperform CS, n erms of numercal effor. In parcular, a he 10 7 level, GBCS needed 25% more compuaonal effor, as depced n Table 2. PEC No. 5 s a fve-componen sysem wh VLE. As expeced, boh CS and GBCS requred sgnfcanly more NFE han he prevously dscussed phase equlbrum problems o be able o converge o he global mnmum of he unconsraned Gbbs free energy funcon, as shown n Fgure 2e. In hs case, he use of GBCS resuled n a reducon of he compuaonal effor rangng from 56% a he 10 3 level o 24% a he 10 7 level, and here s an mprovemen n algorhm relably (see resuls repored n Tables 2 and 3). On he oher hand, PEC No. 6 s an eghhydrocarbon mxure wh VLE, whch requred more NFE o be solved by boh meaheurscs. However, GBCS ouperformed CS, as n he case of PEC No. 5, as llusraed n Fgure 2f. The reducon n compuaonal effor of GBCS ranged from 77% a he 10 3 level o 13% a he 10 7 level, and hs mproved mehod may offer 100% relably for PEC a 300 eraons (see resuls repored n Table 3). PEC No. 7 s a nne-componen mxure ha shows a VLE, whch s modeled usng he Soave Redlch Kwong equaon of sae (SRK EoS). The reducon of compuaonal effor a all levels when usng GBCS were more han 53% and 75% when compared wh he orgnal CS, as shown n Table 2 and Fgure 2g. Snce hs s a dffcul phase spl calculaon, hs sgnfcan reducon n NFE ranslaes o a consderable dfference n compuaonal me. In addon, he relably of GBCS s 100% a 200 eraons, whle CS dd no converge o he global opmum of Δg/RT a he same numercal effor. Ths same convergence paern s repeaed for PEC No. 8, whch s repored n Fgure 2h. Ths mxure has 10 componens and shows a VLE. In hs case, he reducon n compuaonal effor for GBCS ranged from 83% a he 10 3 level o 26% a he 10 7 level, as depced n Table 2. Agan, GBCS exhbed a success rae of 100% n PEC a 200 eraons, whle CS faled o fnd he global opmum soluon. Fgure 3 shows he global success rae of boh CS and GBCS for esed phase equlbrum problems. I s clear 10832

9 ha GBCS offers a beer performance han CS a early eraons, especally for mulcomponen sysems. Fgure 2 shows ha, for he phase-spl problems wh a lesser number of componens (.e., decson varables), boh CS and GBCS behaved smlarly. However, for problems wh a greaer number of componens (.e., c 5), whch are relavely more dffcul o solve, GBCS sgnfcanly ouperformed CS n all phase equlbrum problems. Ths numercal performance s llusraed n he relably performance profle gven n Fgure 4 and he effcency performance profle of Fgure 5. Boh performance profles show ha GBCS s more effecve han CS for he unconsraned Gbbs free energy mnmzaon, especally n mulcomponen sysems. In summary, our resuls showed ha he ncorporaon of he graden of he unconsraned Gbbs free energy funcon s useful o mprove he radeoff of dversfcaon and nensfcaon sages of he CS algorhm for phase equlbrum calculaons n nonreacve sysems especally for sysems wh several componens. These resuls are relevan for composonal reservor modelng and chemcal process smulaors, where a grea number of phase equlbrum calculaons s requred for mulcomponen sysems and he compuaonal me s crcal and may represen up o 50% of he smulaon oal CPU me. 27,28 I s clear ha hs conex may lm he applcaon and use of global opmzaon algorhms, ncludng meaheurscs. Therefore, our sudy s a sep forward oward speedng up he convergence performance of CS whou compromsng s relably for he global mnmzaon of he Gbbs free energy, hus enhancng s ncorporaon and use n process smulaors for modelng mulcomponen sysems. 5. CONCLUSIONS In hs sudy, we made use of he graden of he unconsraned Gbbs free energy funcon o mprove he performance of he Cuckoo Search (CS) algorhm for he soluon of phase equlbrum problems. The mproved opmzaon algorhm was evaluaed solvng dffcul phase equlbrum problems. Is performance, n comparson wh he orgnal CS algorhm, has been analyzed. The graden-based Cuckoo Search (GBCS) algorhm ouperformed he CS algorhm, n erms of s relably and effcency n solvng phase equlbrum problems, especally for mulcomponen sysems. Snce CS s a relable sochasc opmzaon mehod for solvng phase equlbrum problems, 18 he modfed verson of CS, proposed n hs sudy, made use of he readly avalable graden of he Gbbs funcon o enhance he performance of CS furher, n performng phase equlbrum calculaons, especally n mulcomponen sysems. Thus, GBCS s an mproved sochasc mehod for solvng hs challengng hermodynamc problem. In summary, hs sudy provdes nsghs on he uly of he graden nformaon o mprove he performance of meaheurscs for solvng global opmzaon problems relaed o he phase equlbrum modelng. AUTHOR INFORMATION Correspondng Auhor *E-mal: percole@homal.com Noes The auhors declare no compeng fnancal neres. REFERENCES (1) Wakeman, W. A.; Saeva, R. P. Numercal soluon of he sohermal, sobarc phase equlbrum problem. Rev. Chem. Eng. 2004, 20, (2) Zhang, H.; Bonlla-Percole, A.; Rangaah, G. P. A revew on global opmzaon mehods for phase equlbrum modelng and calculaons. Open Thermodyn. J. 2011, 5, (3) Teh, Y. S.; Rangaah, G. P. A sudy of equaon-solvng and Gbbs free energy mnmzaon mehods for phase equlbrum calculaons. Chem. Eng. Res. Des. 2002, 80, (4) Jalal, F.; Seader, J. D.; Khalegh, S. Global soluon approaches n equlbrum and sably analyss usng homoopy connuaon n he complex doman. Compu. Chem. Eng. 2008, 32, (5) McDonald, C. M.; Floudas, C. A. Glopeq: A new compuaonal ool for he phase and chemcal equlbrum problem. Compu. Chem. Eng. 1997, 21, (6) Burgos-Solorzano, G. I.; Brennecke, J. F.; Sadherr, M. A. Valdaed compung approach for hgh-pressure chemcal and mulphase equlbrum. Flud Phase Equlb. 2004, 219, (7) Ncha, D. V.; Gomez, S.; Luna, E. Mulphase equlbra calculaon by drec mnmzaon of Gbbs free energy wh a global opmzaon mehod. Compu. Chem. Eng. 2002, 26, (8) Rangaah, G. P. Evaluaon of genec algorhms and smulaed annealng for phase equlbrum and sably problems. Flud Phase Equlb. 2001, , (9) Teh, Y. S.; Rangaah, G. P. Tabu search for global opmzaon of connuous funcons wh applcaon o phase equlbrum calculaons. Compu. Chem. Eng. 2003, 27, (10) Srnvas, M.; Rangaah, G. P. A sudy of dfferenal evoluon and abu search for benchmark, phase equlbrum and phase sably problems. Compu. Chem. Eng. 2007, 31, (11) Srnvas, M.; Rangaah, G. P. Dfferenal Evoluon wh Tabu Ls for Global Opmzaon and Is Applcaon o Phase Equlbrum and Parameer Esmaon Problems. Ind. Eng. Chem. Res. 2007, 46, (12) Bonlla-Percole, A.; Segova-Hernandez, J. G. A comparave sudy of parcle swarm opmzaon and s varans for phase sably and equlbrum calculaons n mulcomponen reacve and nonreacve sysems. Flud Phase Equlb. 2010, 289, (13) Zhang, H.; Fernandez-Vargas, J. A.; Rangaah, G. P.; Bonlla- Percole, A.; Segova-Hernandez, J. G. Evaluaon of negraed dfferenal evoluon and unfed bare-bones parcle swarm opmzaon for phase equlbrum and sably problems. Flud Phase Equlb. 2011, 310, (14) Faeen, S. E. K.; Bonlla-Percole, A.; Rangaah, G. P. Evaluaon of Covarance Marx Adapaon Evoluon Sraegy, Shuffled Complex Evoluon and Frefly Algorhms for Phase Sably, Phase Equlbrum and Chemcal Equlbrum Problems. Chem. Eng. Res. Des. 2012, 90, (15) Fernandez-Vargas, J. A.; Bonlla-Percole, A.; Segova- Hernandez, J. G. An mproved an colony opmzaon mehod and s applcaon for he hermodynamc modelng of phase equlbrum. Flud Phase Equlb. 2013, 353, (16) Bhargava, V.; Faeen, S. E. K.; Bonlla-Percole, A. Cuckoo Search: A new naure-nspred opmzaon mehod for phase equlbrum calculaons. Flud Phase Equlb. 2013, 337, (17) Elnabawy, A. O.; Faeen, S. E. K.; Bonlla-Percole, A. Phase sably analyss and phase equlbrum calculaons n reacve and nonreacve sysems usng charged sysem search algorhms. Ind. Eng. Chem. Res. 2014, 53, (18) Faeen, S. E. K.; Bonlla-Percole, A. On he Effecveness of Naure-Inspred Meaheursc Algorhms for Performng Phase Equlbrum Thermodynamc Calculaons. Sc. World J. 2014, 2014, Arcle ID (19) Bosman, P. A. N. On gradens and hybrd evoluonary algorhms for real-valued mul-objecve opmzaon. IEEE Trans. Evol. Compu. 2012, 16, (20) Whe, W. B.; Johnson, S. M.; Danzng, G. B. Chemcal equlbrum n complex mxures. J. Chem. Phys. 1958, 28,

10 (21) Yang, X. S.; Deb, S. Cuckoo search va Le vy flghs. In Proceedngs of World Congress on Naure & Bologcally Inspred Compung (NABIC 2009), December 9 11, 2009, Combaore, Inda; IEEE Publcaons: New York, 2009; pp (ISBN ). (22) Yang, X. S.; Deb, S. Engneerng Opmsaon by Cuckoo Search. In. J. Mah. Modell. Numer. Opm. 2010, 1, (23) Yang, X. S.; Deb, S. Cuckoo search: Recen advances and applcaons. Neural Compu. Appl. 2014, 24, (24) Faeen, S. E. K.; Bonlla-Percole, A. Graden-Based Cuckoo Search for global opmzaon. Mah. Probl. Eng. 2014, 2014, Arcle ID (25) Fser, I, Jr.; Fser, D.; Fser, I. A comprehensve revew of cuckoo search: Varans and hybrds. In. J. Mah. Modell. Numer. Opm. 2013, 4, (26) Dolan, E. D.; More, J. J. Benchmarkng opmzaon sofware wh performance profles. Mah. Progr., Ser. A 2002, 91, (27) Pefrere, M.; Ncha, D. V. Robus and effcen rus-regon based sably analyss and mulphase flash calculaons. Flud Phase Equlb. 2014, 362, (28) Gagans, V.; Vaross, N. An negraed approach for rapd phase behavor calculaons n composonal modelng. J. Pe. Sc. Eng. 2014, 118,