Prediction of ternary ionexchange equilibrium using artificial neural networks and Law of Mass Action


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1 cta Scietiarum ISSN prited: ISSN olie: Doi: /actascitechol.v34i Predictio of terary ioexchage equilibrium usig artificial eural etworks ad Law of Mass ctio Rafael Lua Seh Caevesi 1*, Elizeu velio Zaella Juior 1, Rodrigo ugusto arella 1, Tiago Dias Martis, Marcos Flávio Pito Moreira 1 ad Edso toio da Silva 1 Curso de Egeharia Química, Uiversidade Estadual do Oeste do Paraá, R. da Faculdade, 645, , Toledo, Paraá, razil. Departameto de Termofluidodiâmica, Faculdade de Egeharia Química, Uiversidade Estadual de Campias, Campias, São Paulo, razil. *uthor for correspodece. STRCT. The Law of Mass ctio geerally models the equilibrium data from io exchage processes. This methodology is rigorous i terms of thermodyamics ad takes ito cosideratio the oidealities i the solid ad aqueous phases. However, the artificial eural etworks may also be employed i the phase equilibrium modelig. I this study, both methodologies were tested to describe the io exchage equilibrium i the biary systems SO 4 NO 3, SO 4 Cl , NO 3 Cl  ad i the terary system SO 4 Cl  NO 3, by MERLITE IR 400 resi as io exchager. Datasets used i curret study were geerated by the applicatio of the Law of Mass ctio i the biary systems. Results showed that i the equilibrium modelig of biary systems both methodologies had a similar performace. However, i the predictio of the terary system equilibrium, the rtificial Neural Networks were ot efficiet. Networks were also traied with the iclusio of terary experimetal data. The Law of Mass ctio i the equilibrium modelig of the terary system was more efficiet tha rtificial Neural Networks i all cases. Keywords: artificial eural etwork, mass actio law, ioexchage. plicação de redes eurais artificiais e da Lei da ção das Massas a predição de equilíbrio de sistemas terários de trocaiôica RESUMO. Os dados de equilíbrio de processos de troca iôica são geralmete modelados pelo emprego de Lei da ção das Massas. Esta metodologia é rigorosa do poto de vista termodiâmico e cosidera as ãoidealidades a fase sólida e a fase aquosa. No etato, as redes eurais artificiais também podem ser empregadas a modelagem de equilíbrio de fases. Neste trabalho, ambas as metodologias foram utilizadas para descrever o equilíbrio a troca iôica os sistemas biários SO 4 NO 3, SO 4 Cl , NO 3 Cl  e o sistema terário SO 4 Cl  NO 3  empregado como trocador iôico a resia MERLITE IR 400. No treiameto da rede foram utilizados os dados gerados pela plicação da Lei da ção das Massa os sistemas biários. Os resultados obtidos mostraram que a modelagem de equilíbrio dos sistemas biários ambas as metodologias apresetaram desempeho semelhate, etretato a predição do equilíbrio do sistema terário as Redes Neurais rtificiais ão foram eficietes. Também foram treiadas redes com a iclusão de dados experimetais terários. Na modelagem do equilíbrio do sistema terário, a Lei da ção das Massas foi mais eficiete que as redes eurais em todos os casos. Palavraschave: redes eurais artificiais, lei da ação das massas, troca iôica. Itroductio Ioic exchage is a highly employed process for the treatmet of effluets with ioic species, the purificatio of pharmacological compouds, i which adsorptio of ioic species occurs i a porous material (such as artificial resis or zeolites) ad followed simultaeously by a desorptio process of other ioic species (already preset i the exchager) i equivalet amouts, accordig to the equatio: z + z z + z (1) ± z ± z ± z ± z S R R S ad represet the io pairs; z is the charge of the ioic species; R is the solid phase ad S the liquid oe. Most idustrial applicatios of the io exchage process use fixedbed colum systems. The solutio that would be treated has several distict ios that compete with oe aother for active sites of the
2 54 Caevesi et al. adsorbet material. ccordig to Tamura (004), the uderstadig ad the predictio of io exchage reactios are required for a better quatitative ad efficiet iterpretatio of io exchage processes. Thermodyamic modelig of the io exchage systems has a very importat role i acquirig essetial iformatio for the project of io exchage separatio systems. The Law of Mass ctio pproaches to describe equilibrium i io exchage systems comprise adsorptio isotherms (I) ad the Law of Mass ctio (LM). However, the formulatio of I models, such as Lagmuir s isotherm, fails to take ito accout the effect of the solutio s io force of the couterio that desorpts the exchager. LM is a stricter approach for the represetatio of data equilibrium i io exchage systems. The Law of Mass ctio is a model foregrouded o the fact that io exchage is a reversible process which, accordig to the equatio, is ruled by a chemical equilibrium that defies the selectivity of the io exchager. The reactio s equilibrium costat (K) may be calculated by the followig (MEHLI et al., 1994): K z γ R m γ S y = m γ y γ z S R () m j is the molality of species j i the liquid phase; y j is the mol fractio of the species j i the solid phase; γ Sj is the coefficiet of the activity of the species j i the solutio; γ Rj is the coefficiet of the activity of the species j i the resi. The parameters of the models of the coefficiets of activity ad the compositio of each phase should be kow so that the equilibrium costat of the Equatio could be calculated. Literature shows several models, such as the DebyeHückel, romley, Pitzer ad Che models, for the calculatio of the coefficiet i liquid phases. However, reliable theoretical formulatios for the calculatio of the coefficiet of the activity of ios i the solid phase do ot exist. Smith ad Woodbur (1978) had origially proposed a solutio to this problem which was later used by several authors (LLEN et al., 1989; OYER et al., 1999; CNEVESI et al., 009; MEHLI et al., 1994; SHLLCROSS et al., 1988) who used Wilso s model for the calculatio of the coefficiet of activity for fluid phases to represet the oidealities i the solid phase uder aalysis. The model s parameter was estimated from equilibrium data. Wilso s model had the advatage that it predicted the behavior of the io exchage terary systems whe the rates of equilibrium costats ad the parameters of the models of the coefficiets of activity for the ios i curret phases were kow. Three chemical reactios of biary exchage may occur i a io exchage terary system, depedig o the three equilibrium costats. I this case, the three equilibrium costats ad the fractio of the three compoets ivolved may be related by the followig equatios: z C C C ( ) ( ) C z z z K = K K (3) x + x + xc = 1 (4) Sice the equilibrium costats are a priori kow, a system of oliear equatios may be obtaied. Two equatios are defied from the equilibrium costat ad Equatio 4. system of equatios is thus available whose ukows are the three compositios of the solid phase which may be calculated by the umerical method for the solutio of oliear systems rtificial Neural Networks importat ad highly relevat alterative for the modelig of idustrial processes is the use of rtificial Neural Networks (NNs). I spite of the fact that it has the highest umber of parameters to be determied, NN is a method that calculates variables i a explicit way, or rather, without the eed of solvig a system of oliear equatios. NNs are beig successfully applied i several areas i the idustry of chemical processes, such as, the solutio of differetial equatios, iterpolatio of GPS data, studies of moo ad multicompoet equilibrium data of adsorptio, predictio of stability of phases, modelig of chicke carcasses coolig ad others (FGUNDESKLEN et al., 007; JH; MDRS, 005; KLSSEN et al., 009; PRKSH et al., 008; SCHMITZ et al., 006; SILV et al., 003; SOUZ et al., 006). NNs, a mathematical model based o the eural system of itelliget orgaisms, are capable of learig from experiece ad idetify logical patters i mathematical sequeces. Neuros i NN are placed i layers: the etrace, the itermediate ad the exit layers. Each euro comprises a mathematical logic structure i which the stimuli captured by the syapses are processed
3 Predictio of terary equilibrium of io exchage 55 through the soma fuctio ad the threshold potetial is represeted by trasferece. Equatio 5 represets the above mathematically: N Yk = f ( wk, jxj) + bk (5) j= 1 w is the syaptic weight; x is the etry stimulus, b is the threshold; f represets the trasferece fuctio; Y is the euros exit. Subscripts k ad j represet respectively the umber of layers ad the stimulus. NN applicatio is divided ito three parts: traiig, validatio ad geeralizatio. Data sets are required for NN traiig so that it may idetify patters betwee the etrace ad exit variables ad adjust the syaptic weights by a optimizatio algorithm. The validatio stage cofers whether NN effectively leared the previous traiig ad the geeralizatio stage is the effective use of the adjusted model to the simulatio of the process uder aalysis. NN performace depeds o several factors, such as the umber of itermediate layers, the umber of euros i each layer ad the fuctio of the trasferece employed. The use of a great umber of euros coverges to more precise resposes, although they may trigger a etwork geeralizatio issue whe ew etries occur. However, if the umber of euros is low, there is a possibility that the respose obtaied is ot sufficietly precise. Curret research compares results of the modelig of io exchage process of the biary systems SO 4 NO 3, SO 4 Cl , NO 3 Cl  ad of the predictio of the terary system SO 4 NO Cl , by LM ad NNs, i the cocetratio 0. N at 98 K, employig the resi MERLITE IR 400 as io exchager ad sodium as couterio. Material ad methods The evaluatio of LM ad NN methodologies was udertake by usig equilibrium data of the biary systems SO 4  NO 3, SO 4 Cl  , NO 3 Cl  ad of the terary system SO 4 NO 3 Cl , both at cocetratio 0. N ad temperature 98 K, obtaied by Smith ad Woodbur (1978). These authors ivestigated the io exchage of these ios i solutio usig the sythetic resi MERLITE IR 400, with capacity for aio exchage of 1.4 eq L 1. Modelig by LM LM was employed for the adjustmet of equilibrium data of the biary systems to obtai the parameters of Wilso s model ad the equilibrium costat for each biary system. romley s model was thus used to calculate the coefficiet of io activity i the solutio, accordig to Equatio 6: zi I logγ i = + Fi 1+ I (6) is the DebyeHuckel Costat; I is the io I m force defied by i z i, with zi as the umber of = i ío i loads. F i is the sum of iteractio parameters defied by Equatio 7. F i ( ) z jzi z j + zi = + m j (7) j I z jz i Term is the parameter of romley s model of the electrolyte formed by the catio j ad the aio i. Table 1 shows rates for the systems uder aalysis. Table 1. rates i the calculatio of the coefficiet of io activities i the solutio. Compoud (kg mol 1 ) Na SO NaCl NaNO Wilso s model was employed to calculate the coefficiet of io activity i the solid phase, by Equatio (8). l γ i = 1 y jλij y jλ ji / ykλ jk (8) j= 1 j= 1 k= 1 Λ ij are Wilso s parameters ad is the umber of ios i the solid phase. The applicatio of LM requires estimates of the parameters of iteractio Λ ij. For biary systems Λ ii =1, with crossed parameters determied as from the experimetal data of equilibrium. Parameters i curret research were estimated with miimum quadratic error, represeted by Equatio ad usig Dowhill Simplex method (NELDER; MED, 1965).
4 56 Caevesi et al. _ comp _exp EXP MOD ( R) ( R) (9) F = X X p = 1 p= 1 ( ) EXP R X p is the fractio of the solid phase obtaied experimetally ad ( X ) MOD R is the fractio of the solid phase calculated by the model. Whe rates of parameter Λ ij are estimated, equilibrium curves of each biary system are produced for later utilizatio i NN traiig ad predictio of terary equilibrium data. Curves were produced takig ito accout the compositio i the iterval [0.1], totalig 100 equilibrium scores for each biary pair. Further, terary equilibrium curve was predicted to solve the oliear equatio system by modified NewtoRapso method whe Wilso s parameters ad equilibrium costats from the aalyzed biary system were take ito accout. Modelig by NNs rtificial Neural Networks were also employed i the modelig of equilibrium data of the biary systems SO 4 NO  3, SO 4 Cl , NO  3 Cl  ad of the terary system SO 4 NO  3 Cl . NNs used a logistic fuctio ad oly oe hidde layer for activatio. I all cases, the umber of euros i the etrace ad itermediate layers varied betwee 4 ad 14 to decrease the rate fuctio represeted by the Equatio. Syaptic weights were determied by the Dowhill Simplex method (NELDER; MED, 1965). iaries of data equilibrium previously produced by LM (100 scores for each system) were used for NNs traiig so that a model adequately represetig the exchage process of each system uder aalysis would be obtaied. I this case NNs etry variables were total cocetratio of the liquid phase (N) ad the compositios of each species; compositios i the solid phase were used as exit variables. Terary system s equilibrium data were predicted by employig the 100 data produced i each biary pair (300 scores i all) for traiig. NNs etry ad exit variables were the same as those used i the etwork traiig for modelig the biary data. However, data were fed as terary data, or rather, the ormal fractio of the metal abset i the biary system was presumed to be equal to zero. Several etwork architectures were tested to obtai a structure with a good performace i the predictio of terary equilibrium based o the target aalysis. P p So that the performace of rtificial Neural Networks i the predictio of the terary system could be improved, other tests were udertake usig the etwork structure which had the best performace i previous tests. Five terary experimetal data were radomly iserted to the data set used previously i NN traiig ad thus cocludig the validity, as has bee doe with other methodologies. Results ad discussio Modelig equilibrium biary data y usig ML for the modelig of biary data, the parameters of the systems SO 4 NO  3, SO 4 Cl  ad NO  3 Cl  were adjusted as from the equilibrium biary data of the systems uder aalysis obtaied by Smith ad Woodbur (1978) ad provided i Table. Table shows equilibrium costats, Wilso s parameters ad the rates of target fuctios obtaied i curret research for the optimizatio of these parameters. It may be verified from Table that parameters estimated by the Law of Mass ctio have differet rates tha those origially obtaied by Smith ad Woodbur (1978). This differece is due to the type of target fuctio used i the two research works. The target fuctio i Smith ad Woodbur (1978) was the coefficiet of selectivity ( λ ) defied by Equatio, whereas i curret research it comprised the miimizatio of error amog the compositios of ios i the resi. λ Z Z y m γ S = mγ S y (10) Table. Parameters estimated by the applicatio of the Law of Mass ctio to biary data. System K eq Λ 1 Smith ad Woodbur  SO 4  NO 3 Parameters of Wilso s equatio Λ SO 4  Cl Cl NO Curret research  SO 4  NO SO 4  Cl Cl NO Figures 1, ad 3 show that ML described i a precise way the experimetal data of biary equilibrium obtaied by Smith ad Woodbur (1978). The Law of Mass ctio methodology was successfully employed by Shallcross et al. (1988),
5 Predictio of terary equilibrium of io exchage 57 Valverde et al. (00) ad Vo ad Shallcross (003) who applied it for the predictio of the biary ad terary systems. Several structures were tested to model the biary data by NNs to obtai the structure that best represeted the equilibrium data aalyzed. Table 3 shows structures that produced the best result for each system ad the respective rates of target fuctios ad absolute average deviatio (D). Table 3. Results from the applicatio of NNs to biary data. System Structure D Target Fuctio (103 )  SO 4  NO SO 4  Cl Cl NO Table 4 presets results from the Law of Mass ctio for each system, coupled to the respective rates of the target fuctio ad relative average deviatios (D). Table 4. Results from the applicatio of LM to biary data. System D Target fuctio  SO 4  NO SO 4  Cl Cl NO Table 3 shows that NNs adequately represet biary equilibrium data sice target fuctio ad DD rates, obtaied from each system, were low. The compariso of the two methodologies showed that both described with precisio the experimetal data of equilibrium, which may be observed i Figures 1, ad 3. However, NN applicatio is more advatageous whe compared to that of ML, sice data of resi compositio may be directly obtaied. This is due to the fact that ML requires the solutio of a oliear system with the ukows N1, i which N is the umber of io species that participate i the exchage. I methods for the solutio of equatio systems, their covergece highly depeds o a good iitial estimate. Predictio of terary equilibrium data Two approaches were employed to predict terary equilibrium data, or rather, solvig the equatio system ad predictig by NNs. Whereas i the former, the adjusted parameters of biary systems were used (Table ), predictio by NNs was doe by equilibrium curves produced by ML applied to the biary data for the traiig of etworks ad experimetal data, as a validatio set. Equivalet fractio i the resi Cl LM NO 3 LM Cl EXP NO 3 EXP Equivalet fractio i the resi Cl RN NO 3 RN Cl EXP NO 3 EXP Equivalet fractio i the solutio Figure 1. Equilibrium curves produced for the iary system Cl   NO 3. Equivalet fractio i the solutio Equivalet fractio i the resi SO 4 LM Cl LM SO 4 EXP Cl EXP Equivalet fractio i the resi SO 4 RN Cl RN SO 4 EXP Cl EXP Equivalet fractio i the solutio Figure. Equilibrium curves produced for the biary system SO 4  Cl . Equivalet fractio i the solutio
6 58 Caevesi et al. Equivalet fractio i the resi SO 4 LM NO 3 LM SO 4 EXP NO 3EXP Figure 4 shows results from ML ad NN (structure 113) modelig. Equivalet fractio i the solutio Equivalet fractio i the resi SO 4 RN NO 3 RN SO 4 EXP NO 3 EXP Equivalet fractio i the solutio Figure 3. Equilibrium curves produced for the biary system SO 4  NO 3. Neural etworks with differet structures were tested with variatios betwee 4 ad 14 i the umber of euros of the etry ad itermediate layers. It has bee verified that NN had the best performace with ad target fuctio equal to 3,953 x For the predictio of experimetal data of terary equilibrium, the etwork with the best target fuctio was used. Table 5 shows the results. Table 5. Validatio results of the etwork131. Experimetal Model Y SO4 Y NO3 Y Cl Y SO4 Y NO3 Y Cl Figure 4. Result from Terary Data Modelig. () ML ad () NN. Other tests were udertake with the additio of experimetal data of terary equilibrium applied to the rtificial Neural Network. Figure 5 shows improvemets i the descriptio of the terary system equilibrium. D rates were equal to 11.55% for NN traied with terary data. Usig oly biary data, modelig by NN preseted D equal to 13.15%. Results i Figure 5 show that NNs failed to describe with precisio the experimetal data of equilibrium of the terary system SO 4 Cl  NO 3. This fact demostrates that the methodology is oefficiet i represetig the equilibrium data of the terary system, due to the fact that the etwork was traied oly for biary equilibrium data.
7 Predictio of terary equilibrium of io exchage 59 Figure 5. Result from Modelig of Terary Data by NN with the additio of 5 data from Terary Equilibrium. Coclusio I curret ivestigatio, the efficiecy of the two methodologies, the Law of Mass ctio ad the rtificial Neural Networks, were compared with regard to the represetatio of data of the biary (SO 4 NO  3, SO 4 Cl  ad NO  3 Cl  ) ad terary (SO 4 Cl  NO 3 ) equilibrium. NNs ad the Law of Mass ctio described with efficiecy the biary equilibrium data which may be represeted from D rates give i Tables ad 4, with close results obtaied by ML ad NNs. NNs did ot reveal a good capacity for the predictio of the terary system although rtificial Neural Networks fed with biary ad terary equilibrium data (D = 11.55) had a better efficiecy tha that traied oly with biary data (D = 13.15). The Law of Mass ctio (D = 10.07) maaged to predict satisfactorily the behavior of the terary system equilibrium. I fact, it was the methodology with the highest efficiecy. Nevertheless, the applicatio of NNs may be a alterative to covetioal modelig sice it calculates explicitly the fractio i phases i equilibrium. ML requires the solutio of oliear equatio system. Refereces LLEN, R. M.; DDISON, P..; DECHPUNY,. H. The characterizatio of biary ad terary io exchage equilibria. The Chemical Egieerig Joural, v. 40,. 3, p , OYER, W. D..; IRD, M. H. I.; NIRDOSH, I. Io exchage equilibria i biary ad terary systems. The Caadia Joural of Chemical Egieerig, v. 77,. 1, p. 998, CNEVESI, R. L. S.; JUNIOR, E.. Z.; MRTINS, T. D.; RELL, R..; MOREIR, M. F. P.; SILV, E.. Modelagem do processo de troca iôica pela lei da ação das massas e redes eurais artificiais. Estudos Tecológicos, v. 5,. 3, p , 009. FGUNDESKLEN, M. R.; FERRI, P.; MRTINS, T. D.; TVRES, C. R. G.; SILV, E.. Equilibrium study of the biary mixture of cadmiumzic ios biosorptio by the Sargassum filipedula species usig adsorptio isotherms models ad eural etwork. iochemical Egieerig Joural, v. 34,., p , 007. JH, S. K.; MDRS, G. Neural etwork modelig of adsorptio equilibria of mixtures i supercritical fluids. Idustrial ad Egieerig Chemistry Research, v. 44,. 17, p , 005. KLSSEN, T.; MRTINS, T. D.; CRDOZOFILHO, L.; SILV, E.. Modelagem do sistema de resfriameto por imersão de carcaças de fragos utilizado redes eurais artificiais. cta Scietiarum. Techology, v. 31,., p , 009. MEHLI, M..; SHLLCROSS, D. C.; STEVENS, G. W. Predictio of multicompoet io exchage equilibria. Chemical Egieerig Sciece, v. 49,. 14, p , NELDER, J..; MED, R. simplex method for fuctio miimizatio. The Computer Joural, v. 7,. 4, p , PRKSH, N.; MNIKNDN, S..; GOVINDRJN, L.; VIJYGOPL, V. Predictio of biosorptio efficiecy for the removal of copper(ii) usig artificial eural etworks. Joural of Hazardous Materials, v. 15,. 3, p , 008. SCHMITZ, J. E.; ZEMP, R. J.; MENDES, M. J. rtificial eural etworks for the solutio of the phase stability problem. Fluid Phase Equilibria, v. 45,. 1, p , 006. SHLLCROSS, D. C.; HERRMNN, C. C.; MCCOY,. J. improved model for the predictio of multicompoet io exchage equilibria. Chemical Egieerig Sciece, v. 43,., p , SILV, L. H. M.; NEITZEL, I.; LIM, E. P. Resolução de um modelo de reator de leito fixo ão adiabático com dispersão axial utilizado redes eurais artificiais. cta Scietiarum. Techology, v. 5,. 1, p , 003. SMITH, R. P.; WOODURN, E. T. Predictio of multicompoet io exchage equilibria for the terary system SO 4  NO 3 Cl  from data of biary systems. IChE Joural, v. 4,. 4, p , SOUZ, E. C..; RIEIRO, S. R..; OTELHO, M. F.; KRUEGER, C. P.; CENTENO, J.. S. Geração de isolihas, com dados obtidos por levatameto GPS/L1L, mediate a técica de Redes Neurais rtificiais. cta Scietiarum. Techology, v. 5,., p. 051, 006. TMUR, H. Theorizatio o ioexchage equilibria: activity of species i D phases. Joural of Colloid ad Iterface Sciece, v. 79,. 1, p. 1, 004. VLVERDE, J. L.; DE LUCS,.; GONZLEZ, M.; RODRIGUEZ, J. F. Equilibrium data for the exchage of
8 60 Caevesi et al. Cu +, Cd +, ad Z + Ios for H + o the catioic exchager amberlite IR10. Joural of Chemical ad Egieerig Data, v. 47,. 3, p , 00. VO,. S.; SHLLCROSS, D. C. Multicompoet io exchage equilibria predictio. Chemical Egieerig Research ad Desig, v. 81,. 10, p , 003. Received o March 17, 010. ccepted o February 1, 011. Licese iformatio: This is a opeaccess article distributed uder the terms of the Creative Commos ttributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.
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