ipolytechiPolytech JournaliPolytech Journal2782-40042782-6341Irkutsk National Research Technical University10.21285/1814-3520-2022-4-626-639ipolytech-647Research ArticleЭНЕРГЕТИКАPOWER ENGINEERINGСтационарное уравнение теплового взрыва в среде с распределенной энергией активации: численное решение и приближенияSteady-state equation of thermal explosion in a distributed activation energy medium: numerical solution and approximationshttps://orcid.org/0000-0003-2309-8461ДонскойИ. Г.DonskoyI. G.

Донской Игорь Геннадьевич, кандидат технических наук, старший научный сотрудник Лаборатории термодинамики

664033, г. Иркутск, ул. Лермонтова, 130

Igor G. Donskoy, Cand. Sci. (Eng.), Senior Researcher of the Laboratory of Thermodynamics

130 Lermontov St., Irkutsk 664033, Russia

donskoy.chem@mail.ruИнститут систем энергетики им. Л.А. Мелентьева СО РАНРоссияMelentiev Energy Systems Institute SB RASRussian Federation202202012023264626639Copyright © Донской И.Г., 20232023Донской И.Г.Donskoy I.G.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://ipolytech.elpub.ru/jour/article/view/647

Цель работы – анализ математической модели теплового взрыва в среде с гауссовым распределением реакционной способности; определение критических значений параметров модели и их зависимости от дисперсии распределения. В работе использовалось численное решение краевых задач для стационарного распределения температуры в реакционной среде (метод прогонки с итерационным уточнением функции-источника, метод половинного деления для нахождения критического значения параметра Франк-Каменецкого). Для использованной разностной схемы исследована сеточная сходимость, показан первый порядок точности при численной оценке критического значения параметра Франк-Каменецкого. Расчеты проводились с точностью до третьей значащей цифры. Численные методы реализованы в виде программ в среде MATLAB. Получены численные приближения для решений уравнения теплового взрыва с распределенной энергией активации в квазистационарном приближении. Показано, что критическое значение параметра Франк-Каменецкого связано с дисперсией распределения и параметром Аррениуса простой приближенной аналитической формулой, которая подтверждается путем сравнения с численными оценками. Зависимость критического значения параметра Франк-Каменецкого от дисперсии оказывается гауссовой, поэтому уже при малых значениях дисперсии распределения реакционная среда становится термически неустойчивой. Расчеты показали, что значительная дисперсия реакционной способности (порядка десятых долей от среднего) может наблюдаться только для химических реакций с низкой чувствительностью к температуре (т.е. с малым тепловым эффектом или с низкой энергией активации). Для несимметричных функций распределения также получены приближенные формулы для критических условий. Проведенный анализ позволяет применять предложенную математическую модель для реагирующих сред с распределенной реакционной способностью (например, природных материалов, полимеров, гетерогенных каталитических систем и т.д.) для оценки их тепловой устойчивости.

This work presents a mathematical model of thermal explosion in a medium described by a Gaussian distribution of reactivity, along with the determination of critical values for model parameters and their dependence on the distribution dispersion. The numerical solution of boundary value problems for steady-state temperature distribution in a reaction medium (a sweep method along with the iterative refinement of a source function, a half-interval method to find the critical value of the Frank-Kamenetskii parameter) was used. The grid convergence was investigated for the used difference scheme; the first order of accuracy was observed as a result of numerical evaluation of the critical value of the Frank-Kamenetskii parameter. Calculations were carried out with accuracy to three decimal places. Numerical methods were implemented as programs in the MATLAB environment. Numerical approximations were obtained for solutions of the thermal explosion equation characterised by distributed activation energy in the quasi-steady-state approximation. It was shown that the critical value of the Frank-Kamenetskii parameter is associated with the dispersion of the distribution and the Arrhenius parameter by a simple approximate analytical formula, confirmed by comparing with numerical estimates. Since the dependence of the critical value of the Frank-Kamenetskii parameter on the dispersion is described by a Gaussian function, the reaction medium becomes thermally unstable even at small values of the distribution dispersion. Calculations showed that a significant dispersion of reactivity (on the order of tenths of the average) can be observed only for chemical reactions characterised by low sensitivity to temperature (i.e. a small heat effect or low activation energy). Approximate formulas for critical conditions were also obtained for asymmetrical distribution functions. The analysis allows the proposed mathematical model to be used for assessing the thermal stability of reactive media having distributed reactivity (for example, natural materials, polymers, heterogeneous catalytic systems, etc.).

тепловой взрывраспределенная энергия активациикритические условияthermal explosiondistributed activation energycritical conditionsРабота выполнена в ИСЭМ СО РАН в рамках проекта государственного задания FWEU- 2021 0005 (регистрационный номер АААА-А21-121012190004-5)ReferencesFrank-Kamenetskii D. A. Diffusion and heat exchange in chemical kinetics. Vol. 2171. Princeton University Press, 1955. 384 р. https://doi.org/10.1515/9781400877195.Frank-Kamenetskii D. A. Diffusion and heat exchange in chemical kinetics. Vol. 2171. Princeton University Press; 1955, 384 р. https://doi.org/10.1515/ 9781400877195.Restuccia F., Fernandez-Anez N., Rein G. Experimental measurement of particle size effects on the selfheating ignition of biomass piles: homogeneous samples of dust and pellets // Fuel. 2019. Vol. 256. P. 115838. https://doi.org/10.1016/j.fuel.2019.115838.Restuccia F., Fernandez-Anez N., Rein G. Experi-mental measurement of particle size effects on the self-heating ignition of biomass piles: Homogeneous samples of dust and pellets. Fuel. 2019;256:115838. https://doi.org/10.1016/j.fuel.2019.115838.Померанцев В. В., Шагалова С. Л., Резник В. А., Кушнаренко В. В. Самовозгорание и взрывы пыли натуральных топлив. Л.: Изд-во «Энергия», 1975. 144 с.Pomerantsev V. V., Shagalova S. L., Reznik V. A., Kushnarenko V. V. Spontaneous combustion and dust explosions of natural fuels. Leningrad: Ehnergiya; 1975, 144 p. (In Russ.).Sun Qia, Jiang Lin, Li Mi, Sun Jinhua. Assessment on thermal hazards of reactive chemicals in industry: state of the art and perspectives // Progress in Energy and Combustion Science. 2020. Vol. 78. P. 100832. https://doi.org/10.1016/j.pecs.2020.100832.Sun Qia, Jiang Lin, Li Mi, Sun Jinhua. Assessment on thermal hazards of reactive chemicals in industry: state of the art and perspectives. Progress in Energy and Com-bustion Science. 2020;78:100832. https://doi.org/10.1016/ j.pecs.2020.100832.Lin Qi, Wang Shugang, Liang Yuntao, Song Shuanglin, Ren Tingxiang. Analytical prediction of coal spontaneous combustion tendency: velocity range with high possibility of self-ignition // Fuel Processing Technology. 2017. Vol. 159. P. 38–47. https://doi.org/10.1016/j.fuproc.2016.09.027.Lin Qi, Wang Shugang, Liang Yuntao, Song Shuanglin, Ren Tingxiang. Analytical prediction of coal spontaneous combustion tendency: velocity range with high possibility of self-ignition. Fuel Processing Technology. 2017;159:38-47. https://doi.org/10.1016/j.fuproc.2016.09.027.Boddington T., Gray P., Harvey D. I. Thermal theory of spontaneous ignition: criticality in bodies of arbitrary shape // Philosophical Transactions of the Royal Society А. 1971. Vol. 270. No. 1207. P. 467–506. https://doi.org/10.1098/rsta.1971.0087.Boddington T., Gray P., Harvey D. I. Thermal theory of spontaneous ignition: criticality in bodies of arbitrary shape. Philosophical Transactions of the Royal Society А. 1971;270(1207):467-506. https://doi.org/10.1098/rsta. 1971.0087.Graham-Eagle J. G., Wake G. C. Theory of thermal explosions with simultaneous parallel reactions. II. The two- and three-dimensional cases and the variational method // Proceedings of the Royal Society A. Mathematical, Physical and Engineering Sciences. 1985. Vol. 401. P. 195–202. https://doi.org/10.1098/rspa.1985.0094.Graham-Eagle J. G., Wake G. C. Theory of thermal explosions with simultaneous parallel reactions. II. The two- and threedimensional cases and the variational method. Proceedings of the Royal Society A. Mathematical, Physical and Engineering Sciences. 1985;401:195-202. https://doi.org/10.1098/rspa.1985.0094.Li Shuicai, Liao Shi-Jun. An analytic approach to solve multiple solutions of a strongly nonlinear problem // Applied Mathematics and Computation. 2005. Vol. 169. Iss. 2. P. 854–865. https://doi.org/10.1016/j.amc.2004.09.066.Li Shuicai, Liao Shi-Jun. An analytic approach to solve multiple solutions of a strongly nonlinear problem. Applied Mathematics and Computation. 2005;169(2):854-865. https://doi.org/10.1016/j.amc.2004.09.066.Дик И. Г., Зурер А. Б. Расчет критических условий теплового взрыва в неоднородной среде методом интегральных соотношений // Инженерно-физический журнал. 1980. Т. 38. № 1. С. 95–98.Dik I. G., Zurer A. B. Calculation of critical conditions of a thermal explosion in an inhomogeneous medium by the method of integral relations. Engineering and Physics journal. 1980;38(1):95-98. (In Russ.).Гонтковская В. Т., Гордополова И. С., Перегудов А. Н. О критических условиях теплового взрыва в распределенных системах с цепными реакциями // Физика горения и взрыва. 1987. № 4. С. 64–68.Gontkovskaya V. T., Gordopolova I. S., Peregudov A. N. On critical conditions of a thermal explosion in dis-tributed systems with chain reactions. Fizika Goreniya i Vzryva. 1987;4:64-68. (In Russ.).Novozhilov V. Critical conditions for conjugate thermal explosion // Combustion Theory and Modelling. 2008. Vol. 12. Iss. 3. P. 433–449. https://doi.org/10.1080/13647830701750939.Novozhilov V. Critical conditions for conjugate thermal explosion. Combustion Theory and Modelling. 2008;12(3): 433-449. https://doi.org/10.1080/13647830701750939.Novozhilov V. Thermal explosion in oscillating ambient conditions // Scientific Reports. 2016. Vol. 6. Р. 29730. https://doi.org/10.1038/srep29730.Novozhilov V. Thermal explosion in oscillating ambient conditions. Scientific Reports. 2016;6:29730. https://doi.org/10.1038/srep29730.Merzhanov A. G., Averson A. E. The present state of the thermal igntion theory: an invited review // Combustion and Flame. 1971. Vol. 16. Iss. 1. P. 89–124. https://doi.org/10.1016/S0010-2180(71)80015-9.Merzhanov A. G., Averson A. E. The present state of the thermal ignition theory: an invited review. Combustion and Flame. 1971;16(1):89-124. https://doi.org/10.1016/S0010-2180(71)80015-9.Merzhanov A. G., Barzykin V. V., Shteinberg A. S., Gontkovskaya V. T. Methodological principles in studying chemical reaction kinetics under conditions of programmed heating // Thermochimica Acta. 1977. Vol. 21. Iss. 3. P. 301–322. https://doi.org/10.1016/0040-6031(77)85001-6.Merzhanov A. G., Barzykin V. V., Shteinberg A. S., Gontkovskaya V. T. Methodological principles in studying chemical reaction kinetics under conditions of pro-grammed heating. Thermochimica Acta. 1977;21(3):301-322. https://doi.org/10.1016/0040-6031(77)85001-6.Мержанов А. Г., Озерковская Н. И., Шкадинский К. Г. Динамика теплового взрыва в послеиндукционный период // Физика горения и взрыва. 1999. Т. 35. № 6. С. 65–70.Merzhanov A. G., Ozerkovskaya N. I., Shkadinskii K. G. Thermal explosion dynamics in the post-induction period. Fizika goreniya i vzryva. 1999;35(6):65-70. (In Russ.).Balakrishnan E., Swift A., Wake G. C. Critical values for some non-class A geometries in thermal ignition theory // Mathematical and Computer Modelling. 1996. Vol. 24. Iss. 8. P. 1–10. https://doi.org/10.1016/0895-7177(96)00133-1.Balakrishnan E., Swift A., Wake G. C. Critical values for some non-class A geometries in thermal ignition theory. Mathematical and Computer Modelling. 1996;24(8):1-10. https://doi.org/10.1016/0895-7177(96)00133-1.Sanchez-Rodriguez D., Farjas J., Roura P. The critical conditions for thermal explosion in a system heated at a constant rate // Combustion and Flame. 2017. Vol. 186. P. 211–219. https://doi.org/10.1016/j.combustflame.2017.08.008.Sanchez-Rodriguez D., Farjas J., Roura P. The critical conditions for thermal explosion in a system heated at a constant rate. Combustion and Flame. 2017;186:211-219. https://doi.org/10.1016/j.combustflame.2017.08.008.Melguizo-Gavilanes J., Boettcher P. A., Mevel R., Shepherd J. E. Numerical study of the transition between slow reaction and ignition in a cylindrical vessel // Combustion and Flame. 2019. Vol. 204. P. 116–136. https://doi.org/10.1016/j.combustflame.2018.12.036.Melguizo-Gavilanes J., Boettcher P. A., Mevel R., Shepherd J. E. Numerical study of the transition between slow reaction and ignition in a cylindrical vessel. Combustion and Flame. 2019;204:116-136. https://doi.org/10.1016/j.combustflame.2018.12.036.Zarubin V. S., Kuvyrkin G. N., Savelyeva I. Y. Variational estimates of the parameters of a thermal explosion of a stationary medium in an arbitrary domain // International Journal of Heat and Mass Transfer. 2019. Vol. 135. P. 614–619. https://doi.org/10.1016/j.ijheatmasstransfer.2019.02.009.Zarubin V. S., Kuvyrkin G. N., Savelyeva I. Y. Variational estimates of the parameters of a thermal explosion of a stationary medium in an arbitrary domain. International Journal of Heat and Mass Transfer. 2019;135:614-619.https://doi.org/10.1016/j.ijheatmasstransfer.2019.02.009.Attetkov A. V., Zarubin V. S., Kuvyrkin G. N. Dual variational form of the model of thermal explosion in a quiescent medium with temperature-dependent thermal conductivity // Russian Journal of Physical Chemistry B. 2018. Vol. 12. Iss. 1. P. 91–97. https://doi.org/10.1134/S1990793118010037.Attetkov A. V., Zarubin V. S., Kuvyrkin G. N. Dual variational form of the model of thermal explosion in a quiescent medium with temperature-dependent thermal conductivity. Russian Journal of Physical Chemistry B. 2018. Vol. 12. No. 1. P. 91–97. https://doi.org/10.1134/ S1990793118010037.Miura K., Maki T. A Simple method for estimating f(E) and k0(E) in the distributed activation energy model // Energy Fuels. 1998. Vol. 12. Iss. 5. P. 864–869. https://doi.org/10.1021/ef970212q.Miura K., Maki T. A Simple method for estimating f(E) and k0(E) in the distributed activation energy model. Energy Fuels. 1998;12(5):864-869. https://doi.org/10.1021/ef970212q.Czajka K., Kisiela A., Moron W., Ferens W., Rybak W. Pyrolysis of solid fuels: thermochemical behaviour, kinetics and compensation effect // Fuel Processing Technology. 2016. Vol. 142. P. 42–53. http://dx.doi.org/10.1016/j.fuproc.2015.09.027.Czajka K., Kisiela A., Moron W., Ferens W., Rybak W. Pyrolysis of solid fuels: thermochemical behaviour, kinetics and compensation effect. Fuel Processing Technology. 2016;142:42-53. http://dx.doi.org/10.1016/j.fuproc.2015. 09.027.Li Mi, Jiang Lin, He Jia-Jia, Sun Jin-Hua. Kinetic triplet determination and modified mechanism function construction for thermo-oxidative degradation of waste polyurethane foam using conventional methods and distributed activation energy model method // Energy. 2019. Vol. 175. P. 1–13. https://doi.org/10.1016/j.energy.2019.03.032.Li Mi, Jiang Lin, He Jia-Jia, Sun Jin-Hua. Kinetic triplet determination and modified mechanism function construction for thermo-oxidative degradation of waste polyurethane foam using conventional methods and distributed activation energy model method. Energy. 2019;175:1-13. https://doi.org/10.1016/j.energy.2019.03.032.Ma Junfang, Liu Jiaxun, Jiang Xiumin, Shen Jun. An improved parallel reaction model applied to coal pyrolysis // Fuel Processing Technology. 2021. Vol. 211. P. 106608. https://doi.org/10.1016/j.fuproc.2020.106608.Ma Junfang, Liu Jiaxun, Jiang Xiumin, Shen Jun. An improved parallel reaction model applied to coal pyrolysis. Fuel Processing Technology. 2021;211:106608. https://doi.org/10.1016/j.fuproc.2020.106608.Varhegyi G., Bobály B., Jakab E., Chen Honggang. Thermogravimetric study of biomass pyrolysis kinetics. A distributed activation energy model with prediction tests // Energy Fuels. 2011. Vol. 25. No. 1. P. 24–32. https://doi.org/10.1021/ef101079r.Varhegyi G., Bobály B., Jakab E., Chen Honggang. Thermogravimetric study of biomass pyrolysis kinetics. A distributed activation energy model with prediction tests. Energy Fuels. 2011;25(1):24-32. https://doi.org/10.1021/ef101079r.Wagh A. Modelling and simulation of biomass fast pyrolysis process: kinetics, reactor, and condenser systems // Western Australia School of Mines: Minerals, Energy and Chemical Engineering. 2019. http://hdl.handle.net/20.500.11937/82905.Wagh A. Modelling and simulation of biomass fast pyrolysis process: kinetics, reactor, and condenser systems. Western Australia School of Mines: Minerals, Energy and Chemical Engineering. 2019. http://hdl.handle.net/20.500.11937/82905.Wichman I. S. On the use of operator-splitting methods for the equations of combustion // Combustion and Flame. 1991. Vol. 83. Iss. 3-4. P. 240–252. https://doi.org/10.1016/0010-2180(91)90072-J.Wichman I. S. On the use of operator-splitting methods for the equations of combustion. Combustion and Flame. 1991;83(3-4):240-252. https://doi.org/10.1016/ 0010-2180(91)90072-J.Мержанов А. Г., Зеликман Е. Г., Абрамов В. Г. Вырожденные режимы теплового взрыва // Доклады Академии наук СССР. 1968. Т. 180. № 3. С. 639–642.Merzhanov A. G., Zelikman E. G., Abramov V. G. De-generated modes of a thermal explosion. Doklady Akademii nauk SSSR. 1968;180(3):639-642. (In Russ.).

The authors declare that there are no conflicts of interest present.