Abstract
Bayesian inference was used to test the powdered cellulose pyrolysis, under the isothermal experimental conditions. A completely new procedure that was based on obtaining the reliable distribution functions of the effective (apparent) activation energy (E a ) values by the statistical derivation of prior and posterior functions was introduced. It has been found that the pyrolysis of the powdered cellulose can be described by the kinetics, which differs from the first-order model. It was established that the apparent activation energy value presented as average magnitude in the conversion fraction range of 0.20 ≤ α ≤ 0.65 does not represent the “lumped” kinetic parameter, so in indicated conversion range, the pyrolysis process can be described through single-step reaction model with six-eighths-order (n * = 0.75) kinetics. Based on the presented Bayesian inference results, it was assumed that mechanism of pyrolysis takes place through the decomposition reactions which start from the cellulose chains. From the main characteristics of the prior distribution, relationship between the ingredients of Bayesian inference and the cellulose characteristic energy constant (c) [which is related to the rigidity angle (ψ) as a measure of tenseness of the cellulose chains] has been established in this paper. Based on evaluated prior and posterior distributions and their characteristics, it was found that the pyrolysis process of powdered cellulose takes place probably through formation of levoglucosan, where depolymerization represents the primary reaction path. Bayesian approach can be applied to highly structured reaction systems and complex physico-chemical processes, which include the reactivity distribution of various energy counterparts, which has been often un-tractable by traditional statistical access.
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References
Agrawal R (1988a) Kinetics of reactions involved in pyrolysis of cellulose I. The three reaction model. Can J Chem Eng 66:403–412
Agrawal R (1988b) Kinetic of reactions involved in pyrolysis of cellulose II. The modified Kilzer-Broido model. Can J Chem Eng 66:413–418
Akinrinola FS, Darvell LI, Jones JM, Williams A, Fuwape JA (2014) Characterization of selected Nigerian biomass for combustion and pyrolysis applications. Energy Fuels 28:3821–3832
Aldrich J (1997) RA fisher and the making of maximum likelihood 1912–1922. Stat Sci 12:162–176
Antal M, Varhegyi G (1995) Cellulose pyrolysis kinetics: the current state of knowledge. Ind Eng Chem Res 34:703–717
Arora S, Lal S, Kumar S, Kumar M, Kumar M (2011) Comparative degradation kinetic studies on three biopolymers: chitin, chitosan and cellulose. Arch Appl Sci Res 3:188–201
Aven T, Kvalǿy JT (2002) Implementing the Bayesian paradigm in risk analysis. Reliab Eng Sys Saf 78:195–201
Balogun AO, Lasode OA, McDonald AG (2014) Thermo-analytical and physico-chemical characterization of woody and non-woody biomass from an agro-ecological zone in Nigeria. Bioresources 9:5099–5113
Bamford CH, Tipper CFH (1980) Comprehensive Chemical kinetics. In: Bamford CH, Tipper CFH (eds.) Vol. 22, Elsevier, Amsterdam
Belgacem MN, Gandini A (2005) Surface modification of cellulose fibres. Polímeros: Ciência e Tecnologia 15:114–121
Bigger S, Scheirs J, Camino G (1998) An investigation of the kinetics of cellulose degradation under non-isothermal conditions. Polym Degrad Stab 62:33–40
Blasi C (1994) Numerical simulation of cellulose pyrolysis. Biomass Bioenerg 7:87–98
Bradbury A, Sakai Y, Shafizadeh F (1979) Kinetic model for pyrolysis of cellulose. J Appl Polym Sci 23:3271–3280
Burchard W (1971) Statistics of stiff chain molecules: III. Chain length dependence of the mean square radius of gyration of cellulose- and amylose-tricarbanilates. British Polym J 3:214–221
Cabrales L, Abidi N (2010) On the thermal degradation of cellulose in cotton fibers. J Therm Anal Calorim 102:485–491
Capart R, Khezami L, Burnham AK (2004) Assessment of various kinetic models for the pyrolysis of a microgranular cellulose. Thermochim Acta 417:79–89
Chen WH, Kuo PC (2011) Isothermal torrefaction kinetics of hemicellulose, cellulose, lignin and xylan using thermogravimetric analysis. Energy 36:6451–6460
Chundawat SPS, Beckham GT, Himmel ME, Dale BE (2011) Deconstruction of lignocellulosic biomass to fuels and chemicals. Ann Rev Chem Biomol Eng 2:121–145
Conesa J, Caballero J, Marcilla A, Font R (1995) Analysis of different kinetic models in the dynamic pyrolysis of cellylose. Thermochim Acta 254:175–192
Diebold J (1994) A unified, global model for the pyrolysis of cellulose. Biomass Bioenerg 7:75–85
Ding HZ, Wang ZD (2008) On the degradation evolution equations of cellulose. Cellulose 15:205–224
Emsley AM, Stevens GC (1994) Kinetics and mechanisms of the low-temperature degradation of cellulose. Cellulose 1:26–56
Eom Y, Kim S, Kim SS, Chung SH (2006) Application of peak property method for estimating apparent kinetic parameters of cellulose pyrolysis reaction. J Ind Eng Chem 12:846–852
Flynn JH (1997) The ‘temperature integral’—its use and abuse. Thermochim Acta 300(1–2):83–92
Friedman HL (1963) Kinetics of thermal degradation of char-foaming plactics from thermogravimetry—application to a phenolic resin. Polym Sci C 6:183–195
García Barneto A, Vila C, Ariza J, Vidal T (2011) Thermogravimetric measurement of amorphous cellulose content in flax fibre and flax pulp. Cellulose 18:17–31
Grønli M, Antal M, Varhegyi GX (1999) Round-robin study of cellulose pyrolysis kinetics by thermogravimetry. Ind Eng Chem Res 38:2238–2244
Hedwall JA (1938) Reaktionsfahigkeit Festen Stoffe, Barth JA: Leipzig
Iyer SK, Manjunath D, Manivasakan R (2002) Bivariate exponential distributions using linear structures. Sank Indian J Stat 64:156–166
Janković B (2014) The pyrolysis process of wood biomass samples under isothermal experimental conditions—energy density considerations: application of the distributed apparent activation energy model with a mixture of distribution functions. Cellulose 21:2285–2314
Jovanović R (1989) Edition: the science of fiber and fiber technology. II. Cellulose natural and chemical fibers. Building Book Press, University of Belgrade, Belgrade, pp 112–118
Karabatsos G, Walker SG (2006) On the normalized maximum likelihood and Bayesian decision theory. J Math Psychol 50:517–520
Khachani M, El Hamidi A, Halim M, Arsalane S (2014) Non-isothermal kinetic and thermodynamic studies of the dehydroxylation process of synthetic calcium hydroxide Ca(OH)2. J Mater Environ Sci 5:615–624
Khawam A, Flanagan DR (2006) Solid-state kinetic models: basics and mathematical fundamentals. J Phys Chem B 110:17315–17328
Kim S, Eom Y (2006) Estimation of kinetic triplet of cellulose pyrolysis reaction from isothermal kinetic results. Korean J Chem Eng 23:409–414
Kloczkowski A, Kolinski A (2007) Theoretical models and simulations of polymer chains. In: Mark JE (ed) Physical properties of polymers handbook. Springer Science + Business Media, New York, pp 67–83
Langston PA, Burbidge AS, Jones TF, Simmons MJH (2001) Particle and droplet size analysis from chord measurements using Bayes theorem. Powder Technol 116:33–42
Lédé J (2012) Cellulose pyrolysis kinetics: an historical review on the existence and role of intermediate active cellulose. J Anal Appl Pyrol 94:17–32
Lee PM (2012) Bayesian statistics: an introduction, 4th edn. Wiley, London, pp 36–77
Lee G, Nowak W, Jaroniec J, Zhang Q, Marszalek PE (2004) Molecular dynamics simulations of forced conformational transitions in 1,6-linked polysaccharides. Biophys J 87:1456–1465
Lesaffre E, Lawson A (2012) Bayesian biostatistics—statistics in practice, part I. Basic concepts in Bayesian methods. Wiley, London, pp 106–114
Lester E, Watts D, Cloke M, Langston P (2003) Determining the composition of binary cial blends using Bayes theorem. Fuel 82:117–125
Liao YF, Wang SR, Ma XQ (2004) Study of reaction mechanisms in cellulose pyrolysis. Prepr Pap Am Chem Soc Div Fuel Chem 49(1):407–411
Liau LCK, Hsieh YP (2005) Kinetic analysis of poly(vinyl butyral)/glass ceramic thermal degradation using non-linear heating functions. Polym Degrad Stab 89:545–552
Liu Z, Jiang Z, Fei B, Liu X (2013) Thermal decomposition characteristics of Chinese fir. Bioresources 8:5014–5024
Luo N, Cao F, Zhao X, Xiao T, Fang D (2007) Thermodynamic analysis of aqueous-reforming of polylols for hydrogen generation. Fuel 86:1727–1736
Martz H, Waller R (1985) Bayesian reliability analysis. Wiley, New York, pp 35–45
MATLAB® codes, http://www.mathworks.com/products/matlab/examples.html, 2014
Mayo DG (2010) An error in the argument from conditionality and sufficiency to the likelihood principle. In: Mayo DG, Spanos A (eds) Error and inference—recent exchanges on experimental reasoning, reliability and the objectivity and rationality of science. Cambridge University Press, Cambridge, pp 305–314
Mazeau K, Wyszomirski M (2012) Modelling of Congo red adsorption on the hydrophobic surface of cellulose using molecular dynamics. Cellulose 19:1495–1506
Mettler MS, Mushrif SH, Paulsen AD, Javadekar AD, Vlachos DG, Dauenhauer PJ (2012) Revealing pyrolysis chemistry for biofuels production: conversion of cellulose to furans and small oxygenates. Energy Environ Sci 5:5414–5424
Milne TA, Brennan AH, Glenn BH (1990) Sourcebook of methods of analysis for biomass and biomass conversion processes. Elsevier, London, pp 51–67
Mui ELK, Cheung WH, Lee VKC, McKay G (2010) Compensation effect during the pyrolysis of tyres and bamboo. Waste Manag 30:821–830
Muller-Hagedorn M, Bockhorn H, Krebs L, Muller U (2003) A comparative kinetic study on the pyrolysis of three different wood species. J Anal Appl Pyrol 68–69:231–249
Myung JI, Navarro DJ, Pitt MA (2006) Model selection by normalized maximum likelihood. J Math Psychol 50:167–179
OriginLab Tutorial, The nonlinear curve fitter (NLFit) (2014) using the origin’s fitting function builder. ©OriginLab Corporation, http://www.originlab.com
Poletto M, Pistor V, Santana RMC, José Zattera A (2012) Materials produced from plant biomass. Part II: evaluation of crystallinity and degradation kinetics of cellulose. Mater Res 15:421–427
Robert C (2001) The Bayesian choice: from decision-theoretic motivations to computational implementation, 2nd edn. Springer, New York, pp 38–43
Rohde CA (2014) Introductory statistical inference with the likelihood function, Chapter 14. Bayesian inference. Springer, New York, pp 167–181. ISBN 978-3-319-10460-7
Royall R (1997) Statistical evidence: a likelihood paradigm. Chapman & Hall/CRC, New York, pp 1–31
Sánchez-Jiménez PE, Pérez-Maqueda LA, Perejón A, Pascual-Cosp J, Benítez-Guerrero M, Criado JM (2011) An improved model for the kinetic description of the thermal degradation of cellulose. Cellulose 18:1487–1498
Sánchez-Jiménez PE, Pérez-Maqueda LA, Perejón A, Criado JM (2013) Generalized master plots as a straightforward approach for determining the kinetic model: the case of cellulose pyrolysis. Thermochim Acta 552:54–59
Sanders EB, Goldsmith AI, Seeman JI (2002) A model that distinguishes the pyrolysis of d-glucose, d-fructose, and sucrose from that of cellulose. Application to the understanding of cigarette smoke formation. J Anal Appl Pyrol 66:29–50
Seshadri V, Westmoreland PR (2012) Concerted reactions and mechanism of glucose pyrolysis and implications for cellulose kinetics. J Phys Chem A 116:11997–12013
Shaik SM, Koh CY, Nicholas Sharratt P, Tan Reginald BH (2013) Influence of acids and alkalis on transglycosylation and β-elimination pathway kinetics during cellulose pyrolysis. Thermochim Acta 566:1–9
Sonobe T, Worasuwannarak N (2008) Kinetic analyses of biomass pyrolysis using the distributed activation energy model. Fuel 87:414–421
Tang J, Zhuang Q (2009) A global sensitivity analysis and Bayesian inference framework for improving the parameter estimation and prediction of a process-based Terrestrial Ecosystem Model. J Geophys Res 114:D15303–D15322
Varhegyi G, Antal M (1989) Kinetics of the thermal decomposition of cellulose, hemicellulose, and sugar cane bagasse. Energy Fuels 3:329–335
Vyazovkin S (1996) A unified approach to kinetic processing of nonisothermal data. Int J Chem Kinet 28:95–101
Vyazovkin S (1997a) Advanced isoconversional method. J Therm Anal Calorim 49:1493–1499
Vyazovkin S (1997b) Evaluation of activation energy of thermally stimulated solid-state reactions under arbitrary variation of temperature. J Comput Chem 18(3):393–402
Vyazovkin S (2001) Modification of the integral isoconversional method to account for variation in the activation energy. J Comput Chem 22(2):178–183
Vyazovkin S, Wight CA (1997) Isothermal and nonisothermal reaction kinetics in solids: in search of ways toward consensus. J Phys Chem A 101:8279–8284
Vyazovkin S, Wight CA (1999) Model-free and model-fitting approaches to kinetic analysis of isothermal and nonisothermal data. Thermochim Acta 340–341:53–68
Williams PT, Besler S (1996) The influence of temperature and heating rate on the pyrolysis of biomass. Renew Energy 7:233–250
Włodarczyk P (2012) Experimental and theoretical studies on mutarotation in supercooled liquid state. PhD Thesis (A thesis submitted for the degree of Philosophiae Doctor), Prof. Paluch, M., Institute of Physics, Uniwersytecka 4, 40-008 Katowice, University of Silesia, Poland, 2012, pp. 63–68
Yang H, Yan R, Chin T, Tee DL, Chen H, Zheng C (2004) Thermogravimetric analysis—fourier transform infrared analysis of palm oil waste pyrolysis. Energy Fuels 18:1814–1821
Zhang Y, Liu C, Xie H (2014) Mechanism studies on β-D-glucopyranose pyrolysis by density functional theory methods. J Anal Appl Pyrol 105:23–34
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This research work was partially supported by the Ministry of Science and Environmental Protection of Serbia under project No. 172015.
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Janković, B. Estimation of the distribution of reactivity for powdered cellulose pyrolysis in isothermal experimental conditions using the Bayesian inference. Cellulose 22, 2283–2303 (2015). https://doi.org/10.1007/s10570-015-0653-8
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DOI: https://doi.org/10.1007/s10570-015-0653-8