Abstract
This paper explores the hyperbolic heat transfer effects in processes involving high heating rates. The behavior of the model is analyzed in detail under different boundary conditions and the circumstances under which a non-Fourier law could be used to describe thermal conduction processes established from physical mathematical analysis. Finally, the model developed here is coupled to a previous population balance framework to predict the bubbling phenomenon that occurs during the fast pyrolysis of biomass. We found that a transient overheating occurs in the central zone of the generated liquid phase due to the high heating rates that take place during that process.
Funding statement: The authors want to thank to the project “Strategy of transformation of the Colombian energy sector in the horizon 2030” funded by the call 788 of Minciencias Scientific Ecosystem, Contract number FP44842-210-2018. The authors also want to extend thanks to the Alliance for Biomass and Sustainability Research – ABISURE, Hermes code 53024, for its support in the realization of this study.
Acknowledgment
We declare that this paper has been submitted with full responsibility, following the due ethical procedure, and that there is no duplicate publication, fraud, plagiarism, or concerns about animal or human experimentation.
The authors declare no competing interests are at stake and there is no conflict of interest with other people or organizations that could inappropriately influence or bias the content of the paper.
References
[1] J. Montoya, et al., Effect of temperature and heating rate on product distribution from the pyrolysis of sugarcane bagasse in a hot plate reactor, J. Anal. Appl. Pyrolysis 123 (2017), 347–363. doi: 10.1016/j.jaap.2016.11.008.Search in Google Scholar
[2] A. Teixeira, Aerosol generation by reactive boiling ejection of molten cellulose, Energy Environ. Sci. 4 (Sept. 2011), no. 10, 4306–4321. doi: 10.1039/c1ee01876k.Search in Google Scholar
[3] M. Shafi and R. Flumerfelt, Initial bubble growth in polymer foam processes, Chem. Eng. Sci. 52 (Feb. 1997), no. 4, 434. doi: 10.1016/S0009-2509.Search in Google Scholar
[4] H.-Y. Kwak and Y. Kim, Homogeneous nucleation and macroscopic growth of gas bubble in organic solutions, Int. J. Heat Mass Transf. 41 (Feb. 1998), 757–767. doi: 10.1016/S0017-9310(97)00182-8.Search in Google Scholar
[5] K. Butler, A Numerical Model for Combustion of Bubbling Thermoplastic Materials in Microgravity, 2002, p. 70.10.6028/NIST.IR.6894Search in Google Scholar
[6] A. Vedavarz, S. Kumar and M. Moallemi, Significance of Non-Fourier Heat Waves in Conduction, J. Heat Transf. 116 (1994), no. 1, 221–224. doi: 10.1115/1.2910859.Search in Google Scholar
[7] M. Thompson, Non-Fourier Effects at High Heat Flux, J. Heat Transf. 95 (1973), no. 2, 284–286. doi: 10.1115/1.3450051.Search in Google Scholar
[8] P. Genske and K. Stephan, Numerical simulation of heat transfer during growth of single vapor bubbles in nucleate boiling, Int. J. Therm. Sci. 45 (2006), 299–309. doi: 10.1016/j.ijthermalsci.2004.07.008.Search in Google Scholar
[9] K. Herwig, Fourier Versus Non-Fourier Heat Conduction in Materials With a Nonhomogeneous Inner Structure, J. Heat Transf. 122 (2000), 363–365. doi: 10.1115/1.521471.Search in Google Scholar
[10] M. Ozisik and D. Tzou, On the Wave Theory in Heat Conduction, J. Heat Transf. 116 (1994), 526–535. doi: 10.1115/1.2910903.Search in Google Scholar
[11] I. Shnaid, Thermodynamical Approach For The Determination Of The Speed Of Heat Propagation In Heat Conduction, Conf. Proc. 2 (1998), 31027855.Search in Google Scholar
[12] P. Antaki, Importance of nonFourier heat conduction in solid-phase reactions, Combust. Flame 112 (1998), no. 3, 131–134. doi: 10.1016/S0010-2180.Search in Google Scholar
[13] M. Balaban, Effect Of Volume Change In Foods On The Temperature And Moisture Content Predictions Of Simultaneous Heat, J. Food Process Eng. 12 (1989), 67–88. doi: 10.1111/j.1745-4530.1990.tb00041.x.Search in Google Scholar
[14] Y. Zhang, D. Tzou and J. Chen, Micro-and Nanoscale Heat Transfer in Femtosecond Laser Processing of Metals, in: (Aug. 2019).Search in Google Scholar
[15] K. Zhukovsky, D. Oskolkov and N. Gubina, Some Exact Solutions to Non-Fourier Heat Equations, Axioms (2018). doi: 10.3390/axioms7030048.Search in Google Scholar
[16] D. Joseph and L. Luigi, Heat waves D, Rev. Mod. Phys. 61 (1989), no. 1, 41–73. doi: 10.1103/RevModPhys.61.41.Search in Google Scholar
[17] C. Cattaneo, in: Sulla Conduzione Del Calore, 3 (1948).Search in Google Scholar
[18] O. Madelung, Phonon-Phonon Interaction: Thermal Properties 7.1 Introduction, in: Introduction to Solid-State Theory, (1978), 314–326.10.1007/978-3-642-61885-7_7Search in Google Scholar
[19] D. Tzou, Damping and Resonance Characteristics of Thermal Waves, ASME Trans. Ser. E J. Appl. Mech. 59 (1992). doi: 10.1115/1.2894054.Search in Google Scholar
[20] Y. Guo, D. Jou and M. Wang, Macroscopic heat transport equations and heat waves in nonequilibrium states, Phys. D Nonlinear Phenom. 342 (2017), 24–31. doi: 10.1016/j.physd.2016.10.005.Search in Google Scholar
[21] C. Dorao, Simulation of thermal disturbances with finite wave speeds using a high order method, J. Comput. Appl. Math. 231 (2009), no. 2, 637–647. doi: 10.1016/j.cam.2009.04.006.Search in Google Scholar
[22] M. Xu and L. Wang, Thermal oscillation and resonance in dual-phase-lagging heat conduction, Int. J. Heat Mass Transf. (2002). doi: 10.1016/S0017-9310(01)00199-5.Search in Google Scholar
[23] S. Olek, Y. Zvirin and E. Elias, Bubble growth predictions by the hyperbolic and parabolic heat conduction equations, Warme-und Stoffiibertrag. 25 (1990), 17–26. doi: 10.1007/BF01592349.Search in Google Scholar
[24] W. Chan, M. Kelbon and B. Krieger, Modelling and experimental verification of physical and chemical processes during pyrolysis of a large biomass particle, Fuel 64 (1985), no. 11, 90364–90367. doi: 10.1016/0016-2361.Search in Google Scholar
[25] O. Boutin, M. Ferrer and J. Lédé, Flash pyrolysis of cellulose pellets submitted to a concentrated radiation: Experiments and modelling, Chem. Eng. Sci. 57 (2002), no. 1, 15–25. doi: 10.1016/S0009-2509(01)00360-8.Search in Google Scholar
[26] A. Sharma, et al., A phenomenological model of the mechanisms of lignocellulosic biomass pyrolysis processes, Comput. Chem. Eng. 60 (2014), 231–241. doi: 10.1016/j.compchemeng.2013.09.008.Search in Google Scholar
[27] R. Anca-Couce and P. Sommersacher, Online experiments and modelling with a detailed reaction scheme of single particle biomass pyrolysis Authors, J. Anal. Appl. Pyrolysis 127 (2017), 411–425. doi: 10.1016/j.jaap.2017.07.008.Search in Google Scholar
[28] J. Montoya, et al., Single particle model for biomass pyrolysis with bubble formation dynamics inside the liquid intermediate and its contribution to aerosol formation by thermal ejection, J. Anal. Appl. Pyrolysis 124 (2017), 204–218. doi: 10.1016/j.jaap.2017.02.004.Search in Google Scholar
[29] J. Montoya, Kinetic Study and Phenomenological Modeling of a Biomass Particle during Fast Pyrolysis Process, 2016.Search in Google Scholar
[30] A. Anca-Couce and N. Zobel, Numerical analysis of a biomass pyrolysis particle model: Solution method optimized for the coupling to reactor models, Fuel 97 (2012), 80–88. doi: 10.1016/j.fuel.2012.02.033.Search in Google Scholar
[31] R. Leveque, High-Resolution Methods, in: Finite Volume Methods for Hyperbolic Problems, (2002), 100–128.10.1017/CBO9780511791253.007Search in Google Scholar
[32] F. Shakeri and M. Dehghan, The method of lines for solution of the onedimensional wave equation subject to an integral conservation condition, Comput. Math. Appl. 56 (2008), no. 9, 2175–2188. doi: 10.1016/j.camwa.2008.03.055.Search in Google Scholar
[33] D. Maillet, A review of the models using the Cattaneo and Vernotte hyperbolic heat equation and their experimental validation, Int. J. Therm. Sci. 139 (May 2019), 424–432. doi: 10.1016/J.IJTHERMALSCI.2019.02.021.Search in Google Scholar
[34] J. Molina, O. M. Rivera and E. Berjano, Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study, Proc. R. Soc. A, Math. Phys. Eng. Sci. 470 (Oct. 2014), no. 2172, 1–16. doi: 10.1098/RSPA.2014.0547.Search in Google Scholar
[35] D. Y. Tzou, Thermal Shock Phenomena Under High Rate Response in Solids, Annu. Rev. Heat Transf. 4 (1992), 111–185.10.1615/AnnualRevHeatTransfer.v4.50Search in Google Scholar
[36] C. Körner and H. W. Bergmann, The physical defects of the hyperbolic heat conduction equation, Appl. Phys. A, Mater. Sci. Process. 67 (Jan. 1998), no. 4, 397–401. doi: 10.1007/s003390050792.Search in Google Scholar
[37] D. Y. Tzou, A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales, J. Heat Transf. 117 (Feb. 1995), no. 1, 8–16. issn: 0022-1481. doi: 10.1115/1.2822329. eprint: https://asmedigitalcollection.asme.org/heattransfer/article-pdf/117/1/8/5790148/8_1.pdf. url: https://doi.org/10.1115/1.2822329.Search in Google Scholar
[38] F. Xu, K. A. Seffen and T. J. Lu, Non-Fourier analysis of skin biothermomechanics, Int. J. Heat Mass Transf. 51 (2008), no. 9, 2237–2259. issn: 0017-9310. doi: https://doi.org/10.1016/j.ijheatmasstransfer.2007.10.024. url: https://www.sciencedirect.com/science/article/pii/S0017931007006552.10.1016/j.ijheatmasstransfer.2007.10.024Search in Google Scholar
[39] R. Zwanzig, Memory Effects in Irreversible Thermodynamics, Phys. Rev. 124 (Nov. 1961), no. 4, 983. doi: 10.1103/PhysRev.124.983.Search in Google Scholar
[40] L. F. Shampine and M. W. Reichelt, The MATLAB ODE Suite, SIAM J. Sci. Comput. 18 (1997), no. 1, 1–22. doi: 10.1137/S1064827594276424. url: 10.1137/S1064827594276424.Search in Google Scholar
[41] J. Montoya, et al., Micro-explosion of liquid intermediates during the fast pyrolysis of sucrose and organosolv lignin, J. Anal. Appl. Pyrolysis 122 (2016), 106–121. issn: 0165-2370. doi: https://doi.org/10.1016/j.jaap.2016.10.010. url: https://www.sciencedirect.com/science/article/pii/S0165237016304053.10.1016/j.jaap.2016.10.010Search in Google Scholar
[42] H. P. Langtangen and A. Logg, Solving PDEs in Python: The FEniCS Tutorial I, 1st ed., Springer Publishing Company, Incorporated, 2017. isbn: 3319524615.10.1007/978-3-319-52462-7Search in Google Scholar
[43] S. Ortiz-Laverde, et al., Proposal of an open-source computational toolbox for solving PDEs in the context of chemical reaction engineering using FEniCS and complementary components, Heliyon 7 (2021), no. 1, e05772. issn: 2405-8440. doi: https://doi.org/10.1016/j.heliyon.2020.e05772. url: https://www.sciencedirect.com/science/article/pii/S2405844020326153.10.1016/j.heliyon.2020.e05772Search in Google Scholar PubMed PubMed Central
[44] W. Kaminski, Hyperbolic Heat Conduction Equation for Materials With a Nonhomogeneous Inner Structure, J. Heat Transf. 112 (Aug. 1990), no. 3, 555–560. issn: 0022-1481. doi: 10.1115/1.2910422. eprint: https://asmedigitalcollection.asme.org/heattransfer/article-pdf/112/3/555/5565742/555_1.pdf. url: https://doi.org/10.1115/1.2910422.Search in Google Scholar
[45] J. A. Lopez Molina, Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study, Proc. R. Soc. A, Math. Phys. Eng. Sci. 470 (2014), no. 2172. issn: 0370-1662. doi: 10.1098/rspa.2014.0547.Search in Google Scholar
[46] J. Montoya, et al., Micro-explosion of liquid intermediates during the fast pyrolysis of sucrose and organosolv lignin, J. Anal. Appl. Pyrolysis.Search in Google Scholar
[47] J. Maya and F. Chejne, Novel model for non catalytic solid-gas reactions with structural changes by chemical reaction and sintering, Chem. Eng. Sci. 142 (Mar. 2016), 258–268. doi: 10.1016/j.ces.2015.11.036.Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
