Error function analysis in nonlinear parameter estimation method for kinetic and equilibrium sorption models

Document Type : Research Paper

Authors

University of Imam Khomeini

Abstract

A robust method for parameter estimation is required in determining the governing model in kinetic and equilibrium sorption modeling. In nonlinear method, parameters are estimated using a trial and error procedure to minimize an error function. In this study the effect of different error functions on estimated parameters of kinetic models (Pseudo First and second order) and equilibrium models (Langmuir and Freundlich) were examined. Best error function was determined using 239 series of kinetic data and 129 series of equilibrium data which are extracted from 100 previous studies. Using various error functions shows a lot of changes in estimated parameters, so that the largest changes for parameters is about 100%. using normalized error method based on experimental data, best error functions respectively are Errsq, Hybrd and Mpsd with average normal error of 3 and 3.4 and 3.6. Ef and Crm methods have the maximum normalized error according to the experimental data.

Keywords

Main Subjects


* Abd Ed-Latif, MM. Elkady,M.F. (2010) Equilibrium isotherms for harmful ions sorption using nano zirconium vanadate ion exchanger, Desalination, 255 (1–3) 21-43.
* Abdul Mujeeb,V. Alikutty,M. Muraleedharan, P. Synthesis,K. (2014) characterization and vanadium (V) sorption studies on some chitosan derivatives, Journal of Water Process Engineering (4),143-148.
* Alistair,B. Boxall,A. Blackwell,P. Cavallo,R. Kay,P. Tolls,J. (2002) The sorption and transport of a sulphonamide antibiotic in soil systems, Toxicology Letters 131(1–2) 19-28.
* Alkan, M. Demirbas¸O¨. z, Celikapa, S. Dogan, M. (2004) Sorption of acid red 57 from aqueous solution onto sepiolite, Journal of Hazardous Materials (116),135–145.
* Arvind, K. Prasad,B. Mishra,I.M.(2014) Adsorption of acrylonitrile from aqueous solution using bagasse fly ash, Journal of Water Process Engineering(2),129-133.
* Benaïssa,H.(2010) Influence of ionic strength on methylene blue removal by sorption from synthetic aqueous solution using almond peel as a sorbent Material bnaissa.jtusci.(4), 40-47.
* Bhatt, R. R., & Shah, B. A. (2015). Sorption studies of heavy metal ions by salicylic acid–formaldehyde–catechol terpolymeric resin: Isotherm, kinetic and thermodynamics. Arabian Journal of Chemistry, 8(3), 414-426.
* Cataldo,S. Gianguzza,A. Pettignano,A. (2014) Sorption of Pd (II) ion by calcium alginate gel beads at different chloride concentrations and pH. A kinetic and equilibrium study, Arabian Journal of Chemistry.
Chen,Ch. (2013) Evaluation of Equilibrium Sorption Isotherm Equations, Chemical Engineering Journal, (7),24-44.
Cheung,C.W. Porter,J. F and Mckay,G.(2001). Sorption kinetic analysis for the removal of cadmium ions from effluents using bone, Wat. Res., 35(3), 605-612.
* Chip, A and Lena, Ma. (2002). Concentration, pH, and Surface Charge Effects on Cadmium and Lead Sorption in Three Tropical Soils,Published in J. Environ. Qual.(31), 581–589.
* Chowdhury, Sh. Saha, P D. (2011). Biosorption kinetics, thermodynamics and isosteric heat of sorption of Cu(II) onto Tamarindus indica seed powder, Colloids and Surfaces B: Biointerfaces(88) 697– 705.
* Dada,A.O. Olalekan,A.P. Olatunya,A.M. DADA,O. (2012) Langmuir, Freundlich, Temkin and Dubinin–Radushkevich Isotherms Studies of Equilibrium Sorption of Zn2+ Unto Phosphoric Acid Modified Rice Husk, 38-45.
* Degs,AL. KHraisheh,Y. M. A. M and Tutunji,M F. (2001) Sorption of lead ions on diatomite and manganese oxides modified diatomite, Water Research, 35 (15) 3724-3728
* Degs,AL.Yahya,S. El-Barghouthia, M. Majeda Khraishehb, I. Walkerc,A. Gavin, M. (2006). Sorption of Zn(II), Pb(II), and Co(II) using natural sorbents:Equilibrium and kinetic studies, Water Research (40),2645 – 2658.
* Deliyanni,E.A. Bakoyannakis,D.N. Zouboulis,A.I. Matis,K.A. (2003)  Sorption of As(V) ions by akagan_eite-type nanocrystals, Chemosphere (50),155–163.
* Escudero, C., Fiol, N., Villaescusa, I., & Bollinger, J. C. (2013). Effect of chromium speciation on its sorption mechanism onto grape stalks entrapped into alginate beads. Arabian Journal of Chemistry, published online.
* Esmaeilian, A. Mahdavi Mazdeh, A. Ghaforian, H. & Liaghat, A. M. (2015) A novel nanopore biopolymer multi adsorbent for simultaneous removal of anionic and cationic mixtures, Desalination and Water Treatment, 53 (8), 2235–2248.
* Fan,Q.H. Tan,X.L. Wang,X.K.  Montavon, G.(2006). Sorption of Eu(III) on Attapulgite Studied by Batch, XPS and EXAFS Techniques. Applied Radiation and Isotopes 67(9) 1600-1608.
* Feng,N.F.Z. (2013) Untreated Chinese ephedra residue as biosorbents for the removal of Pb2+ ions from aqueous solutions, Procedia Environmental Sciences (18), 794 – 799.
* Foo,K.Y. Hameed,B.H. (2010) Insights into the modeling of adsorption isotherm systems, Chemical    Engineering Journal(156),2–10.
* Fytianos, K. Voudrias, E. Kokkalis, E. (2000). Sorption-desorption behavior of 2,4-dichlorophenol by marine sediments, Chemosphere 40(1) 3-6.
* Gürses, A. Karaca, S. Do˘gar, Ç. Bayrak, R. Açıkyıldız, M. and Yalçın, M. (2004). Determination of adsorptive properties of clay/water system: methylene blue sorption, Journal of Colloid and Interface Science, (269),  310–314.
Hana, R. Zhang,J. YuanfengWang, P. Zhao, Zh. Tanga, M. (2009). Study of equilibrium, kinetic and thermodynamic parameters about methylene blue adsorption onto natural zeolite, Chemical Engineering Journal, (145),496–504.
* Hanna, R. uomikoski, S. TeijaKangas, U. Kuokkanen, T. (2014). Chemically activated carbon residue from biomass gasification as a sorbent for iron(II), copper(II) and nickel(II) ions, Journal of Water Process Engineering(4),12–24.
* Ho,Y.S. (2004) Selection of optimum sorption isotherm, Letters to the Editor, Carbon, 42, 2113–2130.
* Ho, Y. S. (2006). Isotherms for the sorption of lead onto peat: comparison of linear and non-linear methods. Polish Journal of Environmental Studies, 15(1), 81-86.
* Ho, Y. S., & Chiang, C. C. (2001). Sorption studies of acid dye by mixed sorbents. Adsorption, 7(2), 139-147.
* Ho, Y. S., & McKay, G. (1998). A comparison of chemisorption kinetic models applied to pollutant removal on various sorbents. Process Safety and Environmental Protection, 76(4), 332-340.
* Ho, Y. S., & McKay, G. (1998). A two-stage batch sorption optimized design for dye removal to minimize contact time. Process Safety and Environmental Protection, 76(4), 313-318.
* Ho, Y. S., & McKay, G. (1999). Pseudo-second order model for sorption processes. Process biochemistry, 34(5), 451-465.
* Ho,Y. S. and Mckay, G. (1999). The sorption of lead (II) ions on peat, Wat. Res., 33(2), 578-584.
Ho,Y.S and McKay, G. (2001) Removal of lead (II) from effluents by sorption on peat using second-order kinetics, Separation science and techlogy, 36 (2) 241–261.
* Ho,Y.S. and McKay, G.(2002) Application of Kinetic Models to the Sorption of Copper(II) on to Peat,  Adsorption Science & Technology (20) No. 8.
* Ho, Y.S. McKay, G. (2003) Sorption of dyes and copper ions onto biosorbents, Process Biochemistry, (38),1047-1061.
* Ho, Y. S., McKay, G., Wase, D. A. J., & Forster, C. F. (2000). Study of the sorption of divalent metal ions on to peat. Adsorption Science & Technology, 18(7), 639-650.
Ho, Y. S., Chiang, C. C., & Hsu, Y. C. (2001). Sorption kinetics for dye removal from aqueous solution using activated clay. Separation Science and Technology, 36(11), 2473-2488.
* Ho,Y.S, Chiu,W.Ta. Wang, Ch. Ch. (2005). Regression analysis for the sorption isotherms of basic dyes on sugarcane dust, Bioresource Technology, 96 (11), 1285-1291.
* Ho,Y.S. Huang, C. T. Huang, H. W. (2002) Equilibrium sorption isotherm for metal ions on tree fern, Process Biochemistry (37),1421_1430.
* Ho,Y.S. Porter,J.F and Mckay, G. (2002). Equilibrium isotherm studies for the sorption of divalent metal ions onto peat : copper, nickel and lead single component systems, Water, Air, and Soil Pollution, (141), 1–33.
Ho, Y. S., and Ofomaja, A. E. (2005). Kinetics and thermodynamics of lead ion sorption on palm kernel fibre from aqueous solution. Process biochemistry, 40(11), 3455-3461.
Ho,Y.S. Ta Chiub, Wen. Hsub, C. S. Huang, C. T. (2004) Sorption of lead ions from aqueous solution using tree fern as a sorbent, Hydrometallurgy (73),55–61.
* Ho,Y.S. Wang, C. C. (2004) Pseudo-isotherms for the sorption of cadmium ion onto tree fern, Process Biochemistry (39), 759–763.
* Jansson,ch. Gubal,M. Rossy,E. Delanche,J.B. and Lecloire,P.(1996) Vanadium(IV) Sorption by chitosan: kinetic and equilibrium, Wat. Res. 30 (2)465-475.
* Javadian, H., Ghorbani, F., Tayebi, H. A., & Asl, S. H. (2015). Study of the adsorption of Cd (II) from aqueous solution using zeolite-based geopolymer, synthesized from coal fly ash; kinetic, isotherm and thermodynamic studies. Arabian Journal of Chemistry, 8(6), 837-849.
* Javier,G. Mar´ıa,M. Joan de, P. Miquel,R. Lara, D. (2007) Arsenic sorption onto natural hematite, magnetite, and goethite, Journal of Hazardous Materials 141, 575–580.
* Jiangxi, Xu. Junfeng,Niu. Zhang,S. (2013) Sorption of perfluorooctane sulfonate (PFOS) on electrospun fiber membranes, Procedia Environmental Sciences (18),472 – 477.
* Khamsa Guechi,El. Hamdaoui,O. (2016) Sorption of malachite green from aqueous solution by potato peel: Kinetics and equilibrium modeling using non-linear analysis method, Arabian Journal of Chemistry, 9(1), 416-424.
* Kumar,M. Tamilarasan,R. (2013) Kinetics, equilibrium data and modeling studies for the sorption of chromium by Prosopis juliflora bark carbon, Arabian Journal of Chemistry, published online.
Lagergern,S. Sven,k.(1898),Comment on “Removal of copper from aqueous solution by aminated and protonated mesoporous aluminas, Journal of Colloid and Interface Science (276), 255–258.
Mahdavi Mazdeh, A. Omid, M.H and Kashefipour, M. (2010). Theoretical and experimental study of linearization methods for second order kinetic equation, Journal of Soil and Water Research, 41(2), 265-272. (In Farsi).
* Mahdavi, A. Esmailian, A. Mokhtari. M. (2009) Comparison of different methods for estimating the parameters of linear and nonlinear absorption equilibrium, Ninth international seminar on river engineering. (In Farsi)
* Moradi, M., Soltanian, M., Pirsaheb, M., Sharafi, K., Soltanian, S., & Mozafari, A. (2014). The Efficiency Study of Pumice Powder to Lead Removal from the Aquatic Environment: Isotherms and Kinetics of the Reaction. Journal of Mazandaran University of Medical Sciences (JMUMS), 1(23), 65-75 (In Farsi).
* Nadeem,M. Mahmood,A. Shahid,S.A. Shah,S.S. Khalid,A.M. and McKay,G. (2006) Sorption of lead from aqueous solution by chemically modified carbon adsorbents Journal of Hazardous Materials138(3) 604-613
* Nagy,B. Maicaneanu,A. Indolean,C. Manzatu,S.D, Carmen,L. Majdik,C,(2013). Comparative study of Cd(II) biosorption on cultivated Agaricus bisporus and wild Lactarius piperatus based biocomposites. Linear and nonlinear equilibrium modelling and kinetics. Journal of the Taiwan Institute of Chemical Engineers 45(3) ,921-929.
* Nagy,B. Szilagyi,B. Majdik,C . Katona,G . Indolean,C and Maicaneanu, Andrada, (2013). Cd (II) and Zn (II) Biosorption on Lactarius piperatus Macrofungus: Equilibrium Isotherm and Kinetic Studies Journal of the Taiwan Institute of Chemical Engineers 45(3) 921-929.
* Nastasovi,A. Sandi,Z. Suruˇci,L.j. Maksin,D. Jakovljevi,O.D.A. (2009)  Kinetics of hexavalent chromium sorption on amino-functionalized macroporous glycidyl methacrylate copolymer, Journal of Hazardous Materials,  (171), 153–159.
* Ncibi,M.C. Mahjoub,B. Seffen,M. Brouers,F. Gaspard,S. (2009). Sorptiondynamic investigation of chromium(VI) onto Posidonia oceanicafibres:Kinetic modelling using new generalized fractal equation, Biochemical Engineering Journal , (46), 141–146.
* Ng,J.C.Y. Cheung,W.H. McKay,G. (2003). Equilibrium studies for the sorption of lead from effluents using chitosan, Chemosphere, (52), 1021–1030.
* Onoja,M. Akpa,E. Unuabonah,I. Small-Sample Corrected Akaike Information Criterion:(2011) An appropriate statistical tool for ranking of adsorption isotherm models, Desalination 272 20–26.
* Özacar,M.A. Sengil,I. (2005) A kinetic study of metal complex dye sorption onto pine sawdust, Process Biochemistry(40),565–572.
* Pathaka,K. Tripathia,S.C. Singhb,K.K. Mahtelea,Manmohan Kumarb,A.K. Gandhi,P.M. (2014) Removal of americium from aqueous nitrate solutions by sorptiononto PC88A Impregnated macroporous polymeric beads, Journal of Hazardous Materials(278), 464–473.
* Quek, S. Y., Wase, D. A. J., & Forster, C. F. (1998). The use of sago waste for the sorption of lead and copper. Water Sa, 24(3), 251-256.
* Reyad,A. Shawabkeha,M. Tutunji,F. (2003). Experimental study and modeling of basic dye sorption by diatomaceous clay, Applied Clay Science, (24), 111 – 120.
* Ruiza, M. Sastreb, A. Guibalc, M. (2000). Eric. Palladium sorption onglutaraldehyde-crosslinked chitosan, Reactive & Functional Polymers(45),155–173.
* Saad,R. Radovic-Hrapovic,Z. Ahvazi,B. Thiboutot,S. Ampleman,G. Hawari,J (2012). Sorption of 2,4-dinitroanisole (DNAN) on lignin, Journal of Environmental Sciences, 24(5) 808–813.
* Saran, A. Kumar,H and Shrivastava,P.(2013) Performance Assessment of Sawdust as Adsorbent for Cr (VI)Removal from Aqueous Solutions: a Kinetic Modeling, Middle-East Journal of Scientific Research 17 (7): 936-940.
* Shahwan,T.(2014). Sorption kinetics: Obtaining a pseudo-second order rate equation based on a mass balance approach, Journal of Environmental Chemical Engineering 2(2), 1001–1006.
* Shin Juang,R and Yin Chung,J.(2004). Equilibrium sorption of heavy metals and phosphate from singleand binary-sorbate solutions on goethite, Journal of Colloid and Interface Science(275), 53–60.
* Shuichi,I. Masanori, H. Bakul,W,(2005) Effects of resin hydrophilicity on water sorption and changes in modulus of elasticity,Biomaterials(26),6449–6459.
* Washington,J.B.R. aida,J.o. ignattelo,jp , (2003). Sorption Hysteresis of Benzene in Charcoal Particles, American Chemical Society.
* Yaneva,Z.L. Koumanova,B.K and Georgieva,N.V. (2012) Linear and Nonlinear Regression Methods for EquilibriumModelling of 𝑝𝑝-Nitrophenol Biosorption by Rhizopus oryzae:Comparison of Error Analysis Criteria, Journal of Chemistry, 2013, 1-11.
* Zafar,I. Khalid,S.  Daud,N.M and Mirza, M.L. (2015) Kinetic Studies of the Adsorption of Thorium Ions onto Rice Husk from Aqueous Media: Linear and Nonlinear Approach, The Nucleus 52, No. 1 14-19.
* Zamouch,M. Hamdaoui,O. (2012). A use of cedar cone for the removal of a cationic dye from aqueous solutions by sorption, Energy Procedia (18), 1047 – 1058.
* Zamouche, M. Hamdaoui,Ou. (2012). Sorption of Rhodamine B by cedar cone: effect of pH and ionic strength, Energy Procedia, (18), 1228 – 1239.
* Zilg, C. Diestche, F. Hoffmann, B. Diertich, C. Mulhaupt, M. Symp, R. (2001). Recent Advances in Polymer Nanocomposites. 16965.