Application of the Genetic Algorithm to Calculate the Interaction Parameters for Multiphase and Multicomponent Systems

Transcription

1 Iran. J. Chem. Chem. Eng. Vol. 26, No.3, 2007 Applcaton of the Genetc Algorthm to Calculate the Interacton Parameters for Multphase and Multcomponent Systems Rashtchan, Davood* + ; Ovays, Saeed; Taghkhan, Vahd; Ghotb, Crous Department of Chemcal and Petroleum Engneerng, Sharf Unversty of Technology, Tehran, I.R. IRAN ABSTRACT: A method based on the Genetc Algorthm (GA) was developed to study the phase behavor of multcomponent and multphase systems. Upon applcaton of the GA to the thermodynamc models whch are commonly used to study the VLE, VLLE and LLE phase equlbra, the physcally meanngful values for the Bnary Interacton Parameters (BIP) of the models were obtaned. Usng the method proposed n ths work the actvty coeffcents for components at nfnte dluton, obtaned from the local composton based models, can be accurately predcted comparng to the expermental data avalable n the lterature. In ths work, a Global Optmzaton Procedure (GOP) based on the GA was developed to obtan the bnary nteracton parameters of the Wlson, NRTL and the UNIQUAC models for a number of systems at varous temperatures. The VLE, VLLE and LLE values of the bnary nteracton parameters for the actvty coeffcents models were compared wth those reported n the lterature for the systems studed n ths work. The results showed that the values reported for the bnary nteracton parameters can predct the actvty coeffcents at nfnte dlutons for the components n the VLE, VLLE and LLE systems studed n ths work. In order to confrm the accuracy of the results, the values for the actvty coeffcents at nfnte dluton n bnary solutons were compared wth those reported n the lterature. The comparson showed that the models studed usng the proposed method can predct the physcally meanngful bnary nteracton parameters among the speces present n solutons. In addton to the accuracy, smplcty, generalty and short CPU computaton tme of the proposed method should be mentoned as some of ts clearest advantages. KEY WORDS: VLE, LLE, VLLE, Mult-component phase behavor, Genetc algorthm. INTRODUCTION Applcaton of the actvty coeffcent models n order to study the phase behavor of the multphase and multcomponent systems at low pressure have been * To whom correspondence should be addressed. + E-mal: rashtchan@sharf.edu /07/3/89 4/$/3.40 commonly used n engneerng practce. For nstance, modelng of the Lqud-Lqud Equlbra (LLE) systems usng the actvty coeffcent models s mportant n 89

2 Iran. J. Chem. Chem. Eng. Rashtchan, D., et al. Vol. 26, No.3, 2007 smulaton of extracton processes. Snce the equatons for modelng of such systems are nonlnear, usually the estmaton of parameters for these systems leads to a nonconvex optmzaton problem. Thus, the multple optma and the multple root problems are the man ptfall n the calculatons. Many dfferent optmzaton technques have been proposed to estmate the adustable parameters for the actvty coeffcent models [-9] used to study the phase behavor of VLE and LLE systems. But most of them suffer from soluton multplcty and exstence of multple roots for BIP s when appled to correlate the expermental data. The technques that use the gradent-based methods, such as Gauss-Newton [], Gauss-Marquardt [2], and sequental quadratc programmng (SQP) methods [3], are conventonal optmzaton technques n the phase equlbrum calculatons. Snce the evaluaton of gradents s one of the erroneous parts of these technques, the smplex pattern search methods have been used to obtan the best values for the BIP of the actvty coeffcent models [4]. However, these methods cannot provde a theoretcal guarantee of global optmalty. In the recent years, new optmzaton technques have been proposed based on the global optmzaton methods [5-9]. Among them the nterval [5-7], branch and bound methods [8] can be mentoned. Due to the roundngerrors produced n the computatons usng these methods, despte the strong theoretcal bass, the results obtaned are not defntely globally optmum results. Another shortcomng s that because of the need to the gradent evaluatons, these methods have a hgher error potental to produce. It should be noted that there are a few technques that use the gradent less random search (RS) methods as the optmzaton engnes [9]. These technques are rarely used to study the phase behavor of VLE, VLLE and LLE systems. In ths work a new technque that s based on the genetc algorthm, was developed to study the phase behavor for a number of VLE, VLLE and LLE systems at varous temperatures. The results for the actvty coeffcents for components at nfnte dluton obtaned n ths work were compared wth those reported n the lterature. GA OVERVIEW GA s an evolutonary algorthm that was frst proposed by Holland [0] and wdely used n the recent decades. GA s have been also used to solve a large varety of the problems n chemcal engneerng. Among them the mult-obectve optmzaton of the Fludzed bed Catalytc Crackng Reactors (FCCR) can be mentoned []. The GA has been also used n optmzaton of polymerzaton reactors [2]. Geyer et al., used an evolutonary algorthm based on the GA for calculaton of the group contrbuton problem n order to predct the thermodynamc propertes for a number of systems [3]. It s a global optmzaton method that searches the soluton space of a functon through the smulaton of the genetc process of bologcal organsms. Snce the GA works wth the ntal populaton whch s generated randomly. Despte most of the optmzaton methods, ths method does not requre the ntal guess to the optmum pont. Because a wde varety of genetc operators have been used n the GA, there are many versons of the GA whch have been developed n solvng the specfc problems. The GA s a drect optmzaton technque and has been used to solve dffcult problems, even wth obectve functons that are not dfferentable. Ths algorthm does not guarantee the fndng of the global optma, but t s generally good to fnd reasonably acceptable solutons to problems. A good overvew of GA has been presented by Beasley [4, 5]. Basc Prncples of GA The frst stage n the standard GA s the generaton of ntal populaton. In the next stages, ths populaton evolves n a cyclc process that conssts of selecton of parents and recombnaton of them to produce new offsprng. The GA operators, manly Crossover and Mutaton, have the responsblty for producng new generatons. Before runnng the GA, we requre a sutable representaton (codng) and a ftness functon whch assgns a fgure of mert to each coded soluton. For any GA, a chromosome or ndvdual representaton s requred. Ths representaton determnes the structure of GA and ts operators. Each chromosome conssts of a number of genes from a certan alphabet. An alphabet could be bnary, floatng pont, matrces, etc. In the frst GA (Holland s desgn), bnary alphabet was used. But Mchalewcz [6] showed that the real-valued 90

3 Iran. J. Chem. Chem. Eng. Applcaton of the Genetc Algorthm Vol. 26, No.3, 2007 codng moves the problem closer to the problem representaton whch offers hgher precson wth more consstent results across replcatons. He also shows that the real-valued GA s an order of magntude more effcent n terms of CPU tme [6]. The selecton of ndvduals to produce successve generatons plays an extremely mportant role n GA. Ths s a probablstc process that the better ndvduals have the hgher probablty for selecton. The most common selecton mechansms are: Roulette wheel selecton and ts extensons, Scalng technques, Tournament, Eltst models, and Rankng methods. Roulette wheel selecton, was the frst selecton mechansm, whch was developed by Holland. However, a good selecton method must prevent fllng the next populaton by the fttest ndvduals of the current populaton, and consequently prevent the premature convergence dffculty. These ponts have been consdered n the scalng and rankng methods. Genetc operators are the basc search mechansms for GA. They are used for creatng new populaton (solutons) from the selected populaton. There are two basc types of these operators: crossover and mutaton. The responsblty of crossover s the recombnaton of two current ndvduals to produce new ndvduals. These new ndvduals may be mutated, usually by a small perturbaton, for more exploraton of the search space. As mentoned above, the applcaton of genetc operators depends on the codng scheme that has been used. For example n the bnary codng scheme, sngle pont crossover, two pont crossover, or unform crossover can be chosen. Whereas the most common crossover mechansms n the floatng pont codng scheme are: smple crossover, arthmetc crossover, and heurstc crossover. Also there are several mutaton mechansms whch have been used n the floatng pont codng scheme, such as: unform mutaton, non-unform mutaton, and boundary mutaton. Ftness functon s ndependent of GA (e.g. stochastc decson rules). Usually t s an obectve functon that returns a numerc value accordng to the fgure of mert of each ndvdual. But n some cases, such as combnatoral optmzaton, the ftness functon s not ust a smple obectve functon. The reproductve cycle of GA contnues untl a termnaton crteron s met. The most frequently used termnaton crteron s a specfed maxmum number of generatons. Another termnaton crteron s the convergence of the populaton to a specfc ndvdual. PARAMETER ESTIMATION In applcaton of thermodynamc models to studyng of the phase equlbrum of multcomponent and multphase systems, t would be qute possble to obtan dfferent sets of adustable parameters for the studed models. Therefore, t seems that exstence of a crteron to udge about the values for the parameters s a must. As a matter of fact, ths crteron should compare the results obtaned from the model wth those reported n the lterature as expermental data. In order to obtan the best values for the parameters of the models studed n ths work, two followng obectve functons were mnmzed usng the expermental data. The obectve functons are as follows: N exp N Yexp, Ycalc, f (a) () Yexp, or N exp N ( Y exp, Ycalc, ) f (a) (2) where n equaton (), Y stands for the actvty coeffcents of components n mxtures, notably n equaton (2), Y denotes the mole fractons of components. The subscrpts exp. and calc. stand for the expermental and calculated values, respectvely. Also n the above equatons,, N and N exp refer to the components, te lne ndces, number of components and number of expermental data respectvely. Smple Procedure The followng well-known soactvty crteron was used to study the phase equlbra for the varous vaporlqud systems at low pressures as: sat x γ p y p (3) wth,2,...,n LP and,2,...,n C where x and y are the mole fractons of the th component n the th lqud phase and the th component n vapor phase respectvely. Also γ s the actvty 2 2 9

4 Iran. J. Chem. Chem. Eng. Rashtchan, D., et al. Vol. 26, No.3, 2007 Fg. : Form of error functon n the VLE system of tertbutanol () and -butanol (2) at 00 mmhg. Lqud phase s modeled by Wlson actvty coeffcent model and the vapor phase s deal. Parameters n Cal/mole. coeffcent of the th component n the th lqud phase whch would be determned at the system temperature. Also sat p s the vapor pressure of th pure component at system temperature. Accordng to equaton (3) the expermental values for the actvty coeffcents can be determned as follows: y.exp p γ,exp (4) x p,exp sat where subscrpt exp stands for the expermental values of γ, x and y. It should be noted that the calculated values of the actvty coeffcents should be determned at the same condtons as the expermental values reported. In ths case the most commonly used form of the obectve functon s defned as follows: N exp LP C N N γk,exp γk,calc f (a) (5) k γk,exp where γ k,exp and γ k,calc are, respectvely, the expermental and calculated values of the actvty coeffcent of k th component n the th lqud phase at the condtons of th experment. The global mnmum of ths obectve functon (Error) would gve the best sets of adustable parameters for thermodynamc models. The complexty of the problem and exstence of multple optma s llustrated n the Fgs. and 2. In these fgures the 2 Fg. 2: The valley that the global and local mnmum ponts exst n the error functon of the VLE system of tert-butanol () and -butanol (2) at 00 mmhg. Lqud phase s modeled by Wlson actvty coeffcent model and the vapor phase s deal. Parameters n Cal/mole. error functons have been plotted versus the varatons n the bnary nteracton parameters of the Wlson actvty coeffcent model for the VLE system of tert-butanol () and -butanol (2) at 00 mmhg. The heavy strp n Fg. s the area where the mnmum ponts are located. Ths strp has been zoomed n Fg. 2. As the color gradually shfts from cyan to red the value of the error functon decreases. Therefore, the pont wth the heavest red color s the global mnmum pont. It s clear that even for these two varable problems, there are many local mnmum ponts, therefore, t s very mportant to use a global optmzaton algorthm to fnd the correct answer. Rgorous Procedure Generally, f the actvty and fugacty coeffcents cannot be determned drectly from the expermental data, the smple procedure s not worthy of use. Parameter estmaton n the systems such as VLE, VLLE at hgh pressure, where the deal gas law s not applcable any longer, are examples of ths case. In such cases the equlbrum condtons take the followng forms: L v x φ y φ (6) wth,2,,n LP and,2,,n c where, V φ and L φ are, respectvely, the fugacty coeffcents of th component n the vapor phase and n the 92

5 Iran. J. Chem. Chem. Eng. Applcaton of the Genetc Algorthm Vol. 26, No.3, 2007 th lqud phase. It s clear that the expermental values of the fugacty coeffcents cannot be determned drectly. Such a dffculty also exsts when two or more lqud phases are n equlbrum wthout exstence of vapor phase. It should be noted that the Lqud-Lqud Equlbrum (LLE) systems are common examples for ths case. In such condtons the equlbrum crtera can be wrtten as: x x2 γ γ (7) 2 Parameter estmaton n all the above cases needs the rgorous procedure. The component mole fractons should be selected as decson varables. The applcable obectve functons for the systems that belong to the rgorous procedure can be wrtten as: N exp NP NC 2(a) y k ( yk,exp k.calc) f (8) where, y k,exp and y k.calc are the expermental and calculated values of mole fracton of k th component n th phase at the condtons of th experment. The best sets of model parameters can be obtaned from the global mnmum of the obectve functon. Because of the exstence of multple roots, the obectve functon n the form of equaton (8) can further complcate the parameter estmaton procedure than that n equaton (5). Implementaton of the Proposed Method Parameter estmaton can be done as an unconstraned (smple procedure) or a constraned (rgorous procedure) optmzaton problem. These can be formulated as forms of equatons (9) and (0), respectvely. Snce the GA can only handle the maxmzaton problems, therefore, n case of mnmzng the obectve functon by the GA t s necessary to maxmze the followng obectve functon as: max :F f(a) (9) max :F f2(a) subect to: Flash Calculatons 2 (0) To ncrease the effcency and accuracy of the optmzaton process, the best ndvduals of each generaton n the GA s consdered as the startng pont of the Nelder-Mead (NM) method and wll progress n a short perod optmzaton by the NM. Floatng pont encodng, ranked selecton, arthmetc crossover and nonunform mutaton were used to mnmze the obectve functon by the GA. The crossover and mutaton operators can be wrtten as the followng equatons respectvely: Chld r Parent + ( r) Parent () Chld ( r) Parent + r Parent 2 and < < IntalPopulaton f r f r t should be noted that: < 0.5 Chld Chld r (Chld Mn ) (2) > 0.5 Chld k k Chld and < k < IntalPopulaton < < Number of Varables k k r (Max k k k Chld where r s a random number that vares between 0 and, Chld and Parent represent the chld and parent chromosomes, respectvely. Frst and second ndces of chld and parent chromosomes are the varable and chromosome numbers, respectvely. Also k s a random number that represents the ndex of a specfed varable for mutaton. In equaton (8) the expermental values of component mole fractons at each phase are avalable, but ther correspondng theoretcal values cannot be drectly calculated and need the conventonal flash calculatons. In ths work for the problems that nvoke the rgorous procedure, our suggested two-ter algorthm that alternately solves the flash calculatons and parameter estmaton problems was used. The flash calculatons provde the theoretcal values of the component mole fractons at each phase. For such problems, each functon evaluaton leads to a number of flash calculatons. Thus, a relable and quck flash algorthm s needed to have reasonable results n terms of accuracy and computaton tme. The flash algorthm proposed by Luca et al., [7] s an approprate algorthm and used n ths work. However, snce the flash problem s a constraned optmzaton wth lnear constrants and bounds on the varables, t s more approprate to use Generalzed Reduced Gradent (GRG) method for such a problem. Thus, the GRG method nstead of the SQP was used n the modfed verson of the flash calculaton algorthm. As a result of ths modfcaton, the same degree of relablty can be k ) 93

6 Iran. J. Chem. Chem. Eng. Rashtchan, D., et al. Vol. 26, No.3, 2007 reached n flash calculatons wth less teraton n comparson wth the method proposed by Luca et al. To determne the upper and lower bound of the bnary nteracton parameters, the values of actvty coeffcents at nfnte dluton could be useful. Accordng to Stadtherr et al., [7] these values are n the nterval [0.03, 09000]. These values were used wth a conservatve vew to determne the boundary values of the bnary nteracton parameters. The values of the ntal populaton can be determned through a tral and error procedure. It should be noted that the reported values correspond to 95% effcency n the result whch means 95% of the dfferent runs led to the same result. RESULTS AND DISCUSSIONS Sx dfferent equlbrum systems have been used to study the performance of the new algorthm developed n ths work. These systems are categorzed nto two groups. The frst group conssts of the three VLE systems at low pressures. To study these systems, smple procedure was used to mnmze the correspondng obectve functon n the form of equaton (5). The second group conssts of one VLLE system at hgh pressures and four LLE systems. For estmatng the parameters of the thermodynamc models that have been used for modelng such systems, t s necessary to use the optmzaton form as equaton (0). The results of the present work have been also assessed by comparng t wth the most wdely used methods n ths feld. These methods have been dscussed n detal elsewhere [7]. In reference [7] the nterval method has been used. Interval method s essentally a global optmzaton method that searches the whole area of the possble soluton by dvdng t to several ntervals and fnally fndng the most optmum maxmum or mnmum soluton. As expounded earler despte ts strong theoretcal background, the nterval method practcally suffers from the roundng error problem and, thus, there s no guarantee to fnd the global optmum pont. However for the two-varable example to be dscussed later there s an opportunty to see the global mnmum pont n advance as shown n Fgs. and 2. Therefore, the global optmalty of the results can be obtaned. Another commonly used method n the area of determnng the bnary nteracton parameters s the Nelder-Mead method whch has been wdely used [20-23]. The Nelder-Mead method s a drect optmzaton method whch suffers from the need to an ntal guess and uses a combnaton of nterpolaton and extrapolaton functons to locate an optmum pont. There s nether a theoretcal nor a practcal proof for ths method to fnd the global optmum pont. Unfortunately, ths method has tradtonally been used to determne the bnary nteracton parameters for the varety of systems (probably because of unavalablty of a global drect optmzaton method). Hence, most of the nteracton parameters reported n the lterature match the local mnmum ponts of the error functon. The results of parameter estmaton for the NRTL, UNIQUAC and the Wlson models obtaned from study of the VLE phase behavor of 2-butanone and hexadecane at 60ºC and of 2-butanone and octadecane at 80 ºC have been presented n the tables and 2, respectvely. Both systems belong to the smple procedure of parameter estmaton. As can be seen from tables and 2, the physcally meanngful results for the actvty coeffcents at nfnte dluton were obtaned and the results are n better consstency wth the expermental data, than those reported n the lterature [8]. The Wlson, NRTL and the UNIQUAC models are respectvely gven through equatons (2) to (5) as follows: ln γ ln(x + Λx ) + (2) Λ x x + Λ wth Λ x v v x Λ + Λ θ exp RT x In equaton (2) whch expresses the Wlson actvty coeffcent model, v and v are the pure lqud molar volumes for components and respectvely. T s the absolute temperature of the system and θ s the energy parameters that need to be estmated. The NRTL actvty coeffcent model takes the followng form: τ ln γ (3) k x x G k G k 94

7 Iran. J. Chem. Chem. Eng. Applcaton of the Genetc Algorthm Vol. 26, No.3, 2007 Table : The results of parameter estmaton for the NRTL, UNIQUAC and the Wlson models obtaned usng the VLE of 2-butanone and Hexadecane at 60 ºC. Expermental data are taken from [8], Expermental wth 4 equlbrum data ponts and 2.79 [9]. Present Work Intal populaton0, Maxmum number of teratons20, CPU tme0.004 s A 2 (Cal/mol) A 2 (Cal/mol) 2 2 γ, Calculated f NRTL UNIQUAC Wlson Ref. [8] A 2 (Cal/mol) A 2 (Cal/mol) 2 2 γ, Calculated f NRTL UNIQUAC Wlson Table 2: The results of parameter estmaton for the NRTL, UNIQUAC and the Wlson models obtaned usng the VLE of 2-butanone and Octadecane at 80 ºC. Expermental data are taken from [8], Expermental wth 9 equlbrum data pont and 2.2 [9]. Intal populaton20, Maxmum number of teratons30, CPU0.0 s A 2 (Cal/mol) A 2 (Cal/mol) 2 2 γ, Calculated f NRTL UNIQUAC Wlson Ref. [8] A 2 (Cal/mol) A 2 (Cal/mol) 2 2 γ, Calculated f NRTL UNIQUAC Wlson

8 Iran. J. Chem. Chem. Eng. Rashtchan, D., et al. Vol. 26, No.3, 2007 τmx mg x G m τ x kg k x kg k k k wth G exp( τ α ) and τ g RT m where α s the nonrandomness factor and g are the energy parameters that agan need to be estmated. The UNIQUAC actvty coeffcent model can be expressed as: Φ Z θ ln γ ln + q ln + L (4) x 2 Φ Φ x L x + q.0 ln Z L (r q ) r 2 θ Φ τ + q x q x r x r x u exp RT θ τ q θ τ θ k k where Z s the coordnaton number and s set to be equal to 0. r and q are the structural parameters for the pure components and u are the energy parameters. As shown n tables 3 and 4, the estmated parameters for the NRTL and the Wlson models usng the VLE data of tert-butanol and -butanol at dfferent pressures correspond to the global mnmum pont of the error functon. It s clear that the reported values n the lterature [20] correspond to the local mnmum pont. The average trend of mnmzng the obectve functon s shown n Fg 3. Snce the best values of each generaton are less than unty whle the average ndvduals are greater than 00, the best values of each generaton could not be shown n Fg. 3, n fact ther representng τ k Fg. 3: The average trend of mnmzng the error functon n the VLE system of tert-butanol () and -butanol (2) at 700 mmhg. Lqud phase s modeled by Wlson and vapor phase s deal. Expermental data s taken from [20]. lne concdes wth the X-axs. Also, these expermental data sets have been used by Stadtherr et al., [7] to estmate the parameters of the Wlson model. They have also obtaned the same parameter values as those reported n the present work. The calculated values of the actvty coeffcents at nfnte dluton for the Wlson model are also reported n table 3. As can be observed n table 5, estmatng the parameters of NRTL model n the LLE system of water, acetc acd and -hexanol has been done at three dfferent cases. In the frst and second cases the nonrandomness factor () s fxed to 0.2 and 0.3, respectvely, whereas n the thrd case these parameters are consdered as decson varables. As shown, the mnmum values of the obectve functon are nearly the same for all three cases, but the case II has the most accurate (optmum) results. Thus the nonrandomness factor of NRTL model does not have an mportant role n the mnmum value of the obectve functon. However, ts effect on the optmum values of parameters s consderable. The frst case (0.2) has also been studed by Al-Muhtasab et al., [2]. From table 5, t s concluded that the results of the present work are more accurate comparng to those reported n the lterature [2]. Snce the new values of the parameters n ths work are dfferent from the ones reported by reference [2] and these parameters lead to a lower value of the obectve functon relatve to results of reference [2], the results of the present work correspond to the new and more optmal soluton. 96

9 Iran. J. Chem. Chem. Eng. Applcaton of the Genetc Algorthm Vol. 26, No.3, 2007 Table 3: The results of parameter estmaton for the Wlson model obtaned usng the VLE data of tert-butanol () and -butanol (2) at four dfferent pressures. The expermental data are taken from [20]. Ref. [20] Ref. [7] Press.(mmHg) /Equlbrum ponts A 2 (Cal/mol) A 2 (Cal/mol) f γ A 2 (Cal/mol) A 2 (Cal/mol) f γ 00/ / / / Intal populaton00, Maxmum number of generatons0 Press.(mmHg) /Equlbrum ponts A 2 (Cal/mol) A 2 (Cal/mol) f γ CPU (s) 00/ / / / Table 4: The results of parameter estmaton for the NRTL model obtaned usng the VLE data of tert-butanol () and -butanol (2) at four dfferent pressures. The expermental data are taken from [20]. Ref. [20] Pressure (mmhg) A 2 (Cal/mol) A 2 (Cal/mol) 2 2 f Intal populaton200, Maxmum number of generatons20 Pressure (mmhg) A 2 (Cal/mol) A 2 (Cal/mol) 2 2 f CPU (s)

10 Iran. J. Chem. Chem. Eng. Rashtchan, D., et al. Vol. 26, No.3, 2007 Table 5: The results of parameter estmaton for the NRTL model obtaned usng the LLE data of water (), acetc acd(2) and -hexanol (3) at 288 K and atm. The expermental data are taken from [2] wth 8 equlbrum data ponts. Ref. [2] Parameters (Cal/mole) f 2 (Int.pop., Max.No.of CPU (s) Generatons) Parameters (Cal/mole) f 2 Case I A A A A A A (00, 0) A A A A A A Case II Case III A A A A A A A A A (00, 0) A A A (200, 20) The values for the parameters of the NRTL and the UNIQUAC models obtaned from the LLE data of water, proponc acd and methyl butyl ketone system are shown n the tables 6 and 7. Usng the values for the parameters presented n tables 6 and 7 the actvty coeffcents of the components at nfnte dluton can be evaluated. Tables 8 and 9 present the values for the parameters of the NRTL and the UNIQUAC models along wth the Root Mean Square Devatons (RMSD) from the expermental data produced from the models usng the LLE data for the system of water, proponc acd and methyl sopropyl ketone at 25, 35 and 45 C. It should be noted that n tables 6 to 9 the equlbrum data reported n reference [22] have been used to obtan the bnary nteracton parameters for the new system whch has not been studed before. The results of the parameter estmaton for the NRTL and the UNIQUAC models for the quaternary LLE system of -octanol (), tert-amyl methyl ether or TAME (2), Water (3) and Methanol (4) at 25 C were reported n tables 0 and, respectvely. As can be seen from tables 0 and, the NRTL and UNIQUAC parameters obtaned usng the method proposed n ths work can be compared wth those reported n the lterature [23]. It should be stated that the nonrandomness factors of the NRTL model were set to be equal to 0.3 for all the components present n solutons. It goes wthout sayng that, because of the hgher dmensonalty, the parameter estmaton for such system could be much more dffcult than the prevous systems wth lower dmensonalty nvokng the rgorous procedure of parameter estmaton. Estmatng the parameters of the Peng-Robnson (PR) equaton of state n the VLLE system of methane (), water (2) and methylcyclohexane (3) at hgh pressure has been consdered by nvokng the rgorous procedure. The results of parameter estmaton for ths system were reported n table 2. The PR equaton of state along wth ts quadratc mxng rules was gven by equaton (5). RT a P (5) v b v(v + b) + b(v b) a b x b x x a a where x stands for the mole fracton for component and a and b are, respectvely, the parameters of pure components calculated usng the crtcal propertes and acenterc factor data. 98

11 Iran. J. Chem. Chem. Eng. Applcaton of the Genetc Algorthm Vol. 26, No.3, 2007 Table 6: The results of parameter estmaton for the NRTL model obtaned usng the LLE system of water (), proponc acd (2) and methyl butyl ketone (3) at atm. The expermental data are taken from [22]. T(ºC)/ No. of Equlbrum ponts 25/2 A A A Parameters (Cal/mole) f 2 (Intal.populaton, Maxmum Number of CPU (s) Generatons) A A A /3 A A A A A A (00, 0) 45/2 A A A A A A Table 7: The results of parameter estmaton for the UNIQUAC model obtaned usng the LLE data of water (), proponc acd (2) and methyl butyl ketone (3) at atm. The expermental data are taken from [22]. T (ºC)/ No. of Equlbrum ponts 25/2 35/3 45/2 A A A A A A A A A Parameters (Cal/mole) f 2 A A A A A A A A A CPU (s) (Intal.populaton, Maxmum Number of Generatons) 80 (00, 0) CONCLUSIONS In ths work the applcaton of the GA for estmaton of the bnary nteracton parameters for the Wlson, NRTL and the UNIQUAC actvty coeffcent models and the PR equaton of state was proposed. To evaluate the performance of ths algorthm, eght dfferent systems at varous physcal condtons were studed. The systems studed here to test the capablty of the GA algorthm are 2-butanone and hexadecane, 2-butanone and octadecane, tert-butanol and -butanol, water, acetc acd and -hexanol, water + proponc acd + methyl butyl ketone, water + proponc acd + methyl sopropyl ketone, -octanol + TAME + water + methanol and methane + water + methylcyclohexane at varous temperatures and pressures. The results obtaned n ths work showed that the hybrd GA can produce physcally meanngful as well as accurate values for the bnary nteracton parameters of the actvty coeffcent models. The values 99

12 Iran. J. Chem. Chem. Eng. Rashtchan, D., et al. Vol. 26, No.3, 2007 Table 8: The results of parameter estmaton for the NRTL model obtaned usng the LLE data of water (), proponc acd (2) and methyl sopropyl ketone (3) at atm. The expermental data are taken from [22]. T(ºC)/ No. of Equlbrum ponts 25/7 35/7 45/7 A A A A A A A A A Parameters (Cal/mole) f 2 (Intal populaton, Maxmum Number CPU (s) of Generatons) A A A A A A A A A (00, 0) Table 9: The results of parameter estmaton for the UNIQUAC model obtaned usng the LLE data of water (), proponc acd (2) and methyl sopropyl ketone (3) at atm. The expermental data are taken from [22]. Temp.(ºC) / No. of Equlbrum ponts 25/7 35/7 45/7 A A A A A A A A A Parameters (Cal/mole) f 2 (Intal populaton, Maxmum Number CPU (s) of Generatons) A A A A A A A A A (00, 0) for the Bnary Interacton Parameters (BIP) were used to obtan the actvty coeffcents for components at nfnte dlutons. The results obtaned from the models along wth ths algorthm were favorably compared wth those reported n the lterature. One mportant ssue that affects the calculated nteracton parameters for the NRTL model s the nonrandomness factor. Because of ts order of magntude, ths non-randomness factor can create some dffcultes durng the calculatons. Therefore, because of the lack of an effcent optmzaton problem to handle the varables wth dfferent orders of magntude, n the technques used n the lterature t s assumed that the value for the nonrandomness factor s consdered to be constant. However, usng a robust method lke the hybrd GA, used n the present work, there s no need to assgn a fxed value to ths factor. In addton to the accuracy, smplcty and the generalty of the new algorthm, short CPU computaton tme should be mentoned as ts clearest advantage. 00

13 Iran. J. Chem. Chem. Eng. Applcaton of the Genetc Algorthm Vol. 26, No.3, 2007 Table 0: The results of parameter estmaton for the NRTL model obtaned usng the LLE data of -Octanol (), TAME (2), Water (3), Methanol (4) at 25 C. Parameters n Cal/mole and 0.3 for all the components. The expermental data are taken from [23] wth 29 equlbrum data ponts. Intal Populaton 00 Maxmum Number of Generatons 0 CPU 88 s A A A A A A A A A A A A f 2 Ref.[23] f A A A A A A A A A A A A Table : The results of parameter estmaton for the UNIQUAC model obtaned usng the LLE data of -Octanol (), TAME (2), Water (3), Methanol (4) at 25 C. Parameters n Cal/mole. The expermental data are taken from [23] wth 29 equlbrum data ponts. Intal Populaton 00 Maxmum Number of Generatons 0 CPU 950 s A A A A A A A A A A A A f 2 Ref.[23] f A A A A A A A A A A A A Table 2: The results of parameter estmaton for the PR model obtaned usng the VLLE data of Methane (), Water (2), Methylcyclohexane (3) at K. The expermental data are taken from [24] wth 4 equlbrum data ponts. Intal Populaton 30. Maxmum Number of Generatons 40. CPU 26 s. K 2 K 2 K 3 K 3 K 23 K 32 f Receved : 22 th Aprl 2006 ; Accepted : 8 th September 2006 REFERENCES [] Anderson, T.F., Abrams, D.S. and Grens, E. A. II, Evaluaton of Parameters for Nonlnear Thermodynamc Models, AIChE J., 24, p. 20 (978). [2] Valko, P. and Vada, S., An Extended Marquardttype Procedure for Fttng Error-n-Varables Models, Comput. Chem. Eng.,, p. 37 (987). [3] Toa, I.B., Begler, L.T., Reduced Successve Quadratc Programmng Strategy for Error-n- Varables Models, Comput. & Chem. Eng. 6, p. 523 (992). [4] Vamos, R. J. and Hass, C. N., Reducton of Ion- Exchange Equlbra Data Usng an Error-n- Varables Approach, AIChE J., 40, p. 556 (994). [5] Moore, R.E., Hansen, E., Leclerc, A., "Recent Advances n Global Optmzaton", Floudas, C.A. and Pardolos, P.M., Eds., Prnceton Unv. Press, Prnceton, NJ, p. 32 (992). [6] Ratz, D., Csendes, T., On the Selecton of Subdvson Drectons n Interval Branch-and-Bound Methods for Global Optmzaton, J. Global Optm., 7, p. 83 (995). [7] Gau, C. Y., Brennecke, J. F. and Stadtherr, M. A., Relable Nonlnear Parameter Estmaton n VLE Modelng, J. Flud Phase Equlbra, 68, p. (2000). [8] Esposto, W. R. and Floudas, C. A., Global Optmzaton n Parameter Estmaton of Non-Lnear Algebrac Models Va the Error-n-Varable Approach, Ind. Eng. Chem. Res., 37, p. 84 (998). [9] Panatescu, G., Int. Chem. Eng. J., 25, p. 692 (985). [0] Holland, J.H., "Adapton n Natural and Artfcal Systems", MIT Press, (975). [] Rahul, B., Kasat, Sanatosh, Gupta, K., Mult- Obectve Optmzaton of an Industral Fludzed- 0

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