mirtrМир транспортаWorld of Transport and Transportation1992-3252Russian University of Transport (RUT)10.30932/1992-3252-2020-18-6-108-117mirtr-2057Research ArticleНАУКА И ТЕХНИКАSCIENCE AND ENGINEERINGОсобенности расчёта характеристик энергетических комплексов, использующих низкопотенциальную энергиюFeatures of Calculating the Characteristics of Energy Complexes Using Low-Grade EnergyДмитренкоА. В.DmitrenkoA. V.

Дмитренко Артур Владимирович – доктор технических наук, профессор кафедры теплоэнергетики железнодорожного транспорта

Москва

Dmitrenko, Artur V. – D.Sc. (Eng), Professor at the Department of Heat Power Engineering of Railway Transport

Moscow

ammsv@yandex.ruКолосоваМ. А.KolosovaM. A.

Колосова Мария Александровна – аспирант кафедры теплоэнергетики железнодорожного транспорта

Москва

Kolosova, Maria A. – Ph.D. student at the Department of Heat Power Engineering of Railway Transport

Moscow

m.a.kolosova@yandex.ruРоссийский университет транспортаРоссияRussian University of TransportRussian Federation202028122020186108117Copyright © Дмитренко А.В., Колосова М.А., 20202020Дмитренко А.В., Колосова М.А.Dmitrenko A.V., Kolosova M.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mirtr.elpub.ru/jour/article/view/2057

Развитие стационарной энергетики представляется важным аспектом внедрения энергосберегающих технологий на транспорте. В России оно обусловлено основными положениями Энергетической стратегии Российской Федерации до 2030 года. В связи с этим актуальной является задача эффективного использования низкопотенциальной теплоты на основе органического цикла Ренкина (ОЦР) в стационарных энергетических комплексах транспорта. В частности, эта задача характерна для котельных, переводимых с мазутного топлива на газ. В этом случае эффективность применения ОЦР в первую очередь будет определяться эффективностью используемых теплообменных аппаратов (ТА) с фазовым переходом, вследствие чего как технически, так и теоретически будет представлять большой интерес задача проектирования и расчёта оптимальных характеристик этих ТА.

В статье представлена расчётно-теоретическая модель переноса тепла при фазовых переходах в турбулентных потоках на основе соотношений, полученных стохастической теорией гидродинамики и теплообмена. Рассматривается моделирование влияния турбулентности при фазовом переходе при неразвитом кипении пузырькового режима. Результаты сравнения показывают удовлетворительное соответствие значений по формуле, полученной на основе стохастических уравнений, со значениями, рассчитанными по эмпирической формуле для течения в трубе, применяемой в инженерной методике проектирования теплообменных аппаратов. Полученные результаты открывают перспективу исследования процессов переноса тепла при фазовых переходах в турбулентных потоках ТА с целью уменьшения их габаритно-массовых характеристик, а также роста энергетической эффективности как самих аппаратов, так эффективности всего энергетического комплекса.

The development of stationary energy seems to be an important aspect of introduction of energy-saving technologies in transportation sector. In Russia, it is conditioned by the main provisions of the Energy Strategy of the Russian Federation until 2030. In this regard, the problem of efficient use of low-grade heat based on the organic Rankine cycle (ORC) in stationary heat energy supply units in the transport industry is urgent. In particular, this task is typical for boiler houses converted from heavy fuel oil to gas fuel. In this case, the efficiency of ORC application will primarily be determined by the efficiency of the used heat exchangers (HE) with a phase transition, as a result of which, both technically and theoretically, the problem of designing and calculating the optimal characteristics of these HE will be of great interest.

The article presents a theoretical and computational model of heat transfer during phase transitions in turbulent flows based on the relations obtained by the stochastic theory of hydrodynamics and heat transfer. The modelling of the effect of turbulence during the phase transition with undeveloped boiling of the bubble mode is considered. The comparison results show satisfactory conformity of the values obtained according to the formula based on stochastic equations with the values calculated according to the empirical formula for the flow in a pipe, used in the engineering method of designing heat exchangers. The results obtained open the prospect for studying the processes of heat transfer during phase transitions in turbulent flows of HE to reduce their overall and mass characteristics, as well as to increase the energy efficiency of both the devices themselves and the efficiency of the entire energy complex.

транспортэнергетический комплекстеплопередачанеразвитое кипениецикл Ренкинастохастические уравненияпузырьковый режимtransportenergy complexheat transferundeveloped boilingRankine cyclestochastic equationsbubble modeReferencesKalinin, E. K., Dreitser, G. A., Kopp, I. Z., Myakochin, A. S. Efficient Surfaces for Heat Exchangers. Fundamentals and Design. New-York, 2002, 392 p.Kalinin, E. K., Dreitser, G. A., Kopp, I. Z., Myakochin, A. S. Efficient Surfaces for Heat Exchangers. Fundamentals and Design. New-York, 2002, 392 p.Алифанов О. М., Артюхин Е. А., Ненарокомов А. В. Обратные задачи в исследовании сложного теплообмена. – М.: Янус-К, 2009. – 300 c.Alifanov, O. M., Artyukhin, E. A., Nenarokomov, A. V. Inverse problems in the study of complex heat exchange [Obratnie zadachi v issledovanii slozhnogo teploobmena]. Moscow, Yanus-K publ., 2009, 300 p.Ландау Л. Д. К проблеме турбулентности // ДАН СССР. – 1944. – Т. 44. – № 8. – С. 339–342.Landau, L. D. On the problem of turbulence [O probleme turbulentnosti]. DAN SSSR, 1944, Vol. 44, Iss. 8, pp. 339–342.Колмогоров А. Н. Новый метрический инвариант транзитивных динамических систем и автоморфизмов пространств Лебега // ДАН СССР. – 1958. – Т. 119. – № 5. – С. 861–864.Kolmogorov, A. N. A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces [Novie metricheskiy invariant tranzitivnykh dinamicheskikh system i avtomorfizmov prostranstv Lebega]. DAN SSSR, 1958, Vol. 119, Iss. 5, pp. 861–864.Колмогоров А. Н. Об энтропии на единицу времени как метрическом инварианте автоморфизмов // ДАН СССР. – 1959. – Т. 124. – № 4. – С. 754–755.Kolmogorov, A. N. On entropy per unit time as a metric invariant of automorphisms [Ob entropii na edinitsu vremeni kak metricheskom invariante avtomorfizmov]. DAN SSSR, 1959, Vol. 124, Iss. 4, pp. 754–755.Колмогоров А. Н. Математические модели турбулентного движения несжимаемой вязкой жидкости // УМН. – 2004. – Т. 59. – Вып. 1 (355). – С. 5–10.Kolmogorov, A. N. Mathematical models of turbulent motion of an incompressible viscous fluid [Matematicheskie modeli turbulentnogo dvizheniya neszhimaemoi vyazkoi zhidkosti]. UMN, 2004, Vol. 59, Iss. 1 (355), pp. 5–10.Lorenz, E. N. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 1963, Vol. 20, pp. 130–141. DOI: https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. Доступ 19.10.2020.Lorenz, E. N. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 1963, Vol. 20, pp. 130–141. DOI: h t t p s : / / d o i .o r g / 1 0 .1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. Last accessed 19.10.2020.Ruelle, D., Takens, F. On the nature of turbulence. Communications in Mathematical Physics, 1971, Vol. 20, pp. 167–192. DOI: https://doi.org/10.1007/BF01646553. (Erratum Vol. 23, pp. 343–344. https://doi.org/10.1007/BF01893621). Доступ 19.10.2020.Ruelle, D., Takens, F. On the nature of turbulence. Communications in Mathematical Physics, 1971, Vol. 20, pp. 167–192. DOI: https://doi.org/10.1007/BF01646553. (Erratum Vol. 23, pp. 343–344. https://doi.org/10.1007/BF01893621). Last accessed 19.10.2020.Feigenbaum, M. The transition to aperiodic behavior in turbulent systems. Communications in Mathematical Physics, 1980, Vol. 77, pp. 65–86. DOI: https://doi.org/10.1007/BF01205039). Доступ 19.10.2020.Feigenbaum, M. The transition to aperiodic behavior in turbulent systems. Communications in Mathematical Physics, 1980, Vol. 77, Iss. 1, pp. 65–86. DOI: https://doi.org/10.1007/BF01205039. Last accessed 19.10.2020.Рабинович М. И. Стохастические автоколебания и турбулентность // УФН. – 1978. – Т. 125. – № 1. – С. 123–168.Rabinovich, M. I. Stochastic self-oscillations and turbulence [Stokhasticheskie avtokolebaniya i turbulentnost’]. UFN, 1978, Vol. 125, Iss. 1, pp. 123–168.Монин А. С. О природе турбулентности // УФН. – 1978. – Т. 125. – № 1. – С. 97–122.Monin, A. S. On the nature of turbulence [O prirode turbulentnosti]. UFN, 1978, Vol. 125, Iss. 1, pp. 97–122.Рабинович М. И., Сущик М. М. Когерентные структуры в турбулентных течениях. Нелинейные волны. Самоорганизация / Под ред. А. В. Гапонова и М. И. Рабиновича. – М.: Наука, 1983. – С. 58–84.Rabinovich, M. I., Sushchik, M. M. Coherent structures in turbulent flows. Nonlinear waves. Selforganization [Kogerentnie struktury v turbulentnykh techeniyakh. Nelineinie volny. Samoorganizatsiya]. Eds. by A. V. Gaponov and M. I. Rabinovich. Moscow, Nauka publ., 1983, pp. 58–84.Заславский Г. М. Стохастичность динамических систем. – М.: Наука, 1984. – 272 c.Zaslavsky, G. M. Stochasticity of dynamical systems [Stokhastichnost’ dinamicheskikh system]. Moscow, Nauka publ., 1984, 272 p.Струминский В. В. Возникновение турбулентности // ДАН СССР. – 1989. – Т. 307. – № 3. – С. 564– 567.Struminsky, V. V. The emergence of turbulence [Vozniknovenie turbulentnosti]. DAN SSSR, 1989, Vol. 307, Iss. 3, pp. 564–567.Самарский А. А., Мажукин В. И., Матус П. П., Михайлик И. А. Z/2‑консервативные схемы для уравнения Кортевега–де Фриса // ДАН. – 1997. – Т. 357. – № 4. – С. 458–461.Samarsky, A. A., Mazhukin, V. I., Matus, P. P., Mikhailik, I. A. Z/2-conservative schemes for the Korteweg–de Vries equation [Z/2-konservativnie skhemy dlya uravneniya Kortevega–de Frisa]. DAN, 1997, Vol. 357, Iss. 4, pp. 458–461.Климонтович Ю. Л. Проблемы статистической теории открытых систем: критерии относительной степени упорядоченности состояний в процессах само- организации // УФН. – 1989. – Т. 158. – Вып. 1. – С. 59–91. DOI: 10.3367/UFNr.0158.198905b.0059. Доступ 19.10.2020.Klimontovich, Yu. L. Problems of the statistical theory of open systems: criteria for the relative degree of ordering of states in self-organization processes [Problemy statisticheskoi teorii otkrytykh system: kriterii otnositelnoi stepeni uporyadochennosti sostoyanii v protsessakh samoorganizatsii]. UFN, 1989, Vol. 158, Iss. 1, pp. 59–91 DOI: 10.3367/UFNr.0158.198905b.0059. Last accessed 19.10.2020.Sreenivasan, К. R. Fractals and multifractals in fluid turbulence. Ann. Rev. FluidMech, 1991, Vol. 23, pp. 539–600.Sreenivasan, К. R. Fractals and multifractals in fluid turbulence. Ann. Rev. FluidMech, 1991, Vol. 23, pp. 539–600.Orzag, S. A., Kells, L. C. Transition to turbulence in plane Poiseuille and plane Couette flow. Journal of Fluid Mechanics, 1980, Vol. 96 (1), pp. 159–205. DOI:10.1017/S0022112080002066. Доступ 19.10.2020.Orzag, S. A., Kells, L. C. Transition to turbulence in plane Poiseuille and plane Couette flow. Couette flow. Journal of Fluid Mechanics, 1980, Vol. 96 (1), pp. 159–205. DOI:10.1017/S0022112080002066. Last accessed 19.10.2020.Priymak, V. G. Splitting dynamics of coherent structures in a transitional round-pipe flow. Dokl. Phys., 2013, Vol. 58, Iss. 10, pp. 457–465.Priymak, V. G. Splitting dynamics of coherent structures in a transitional round-pipe flow. Dokl. Phys., 2013, Vol. 58, Iss. 10, pp. 457–465.Фурсиков А. В. Моментная теория для уравнений Навье – Стокса со случайной правой частью // Изв. РАН. Сер. матем. – 1992. – Т. 56. – № 6. – С. 1273–1315.Fursikov, A. V. Moment theory for the Navier– Stokes equations with a random right-hand side [Momentnaya teoriya dlya uravnenii Navie–Stoksa so sluchainoi pravoi chastiyu]. Izv. RAS. Ser. mat., 1992, Vol. 56, Iss. 6, pp. 1273–1315.Энергетическая стратегия России на период до 2030 года. Распоряжение Правительства Российской Федерации от 13 ноября 2009 г. № 1715‑р. – М., 2009. –144 с.Energy strategy of Russia for the period up to 2030. Order of the Government of the Russian Federation dated November 13, 2009 No. 1715-r [Energeticheskaya strategiya Rossii na period do 2030 goda. Rasporyazhenie Pravitelstva Rossiiskoi Federatsii ot 13 noyabrya 2009 № 1715-r]. Мoscow, 2009, 144 p.Энергетическая стратегия России до 2035 года. Корректировка. – М., 2014. – 25 с.Energy Strategy of Russia until 2035. Correction. [Energeticheskaya strategiya Rossii do 2035 goda. Korrektirovka]. Moscow, 2014, 25 p.Energy Use, Loss and Opportunities Analysis: US Manufacturing and Mining. US Department of Energy Industrial Technologies Program, 2004, 165 p. [Электрон- ный ресурс]: https://www.energy.gov/sites/prod/files/2013/11/f4/energy_use_loss_opportunities_analysis.pdf. Доступ 19.10.2020.Energy Use, Loss and Opportunities Analysis: US Manufacturing and Mining. US Department of Energy Industrial Technologies Program, 2004, 165 p. [Electronic resource]: https://www.energy.gov/sites/prod/files/2013/11/f4/energy_use_loss_opportunities_analysis.pdf. Last accessed 19.10.2020.Chudnovsky, Y., Gotovsky, M., Greenman, M. [et al]. Integrated Steam/Organic Rankine Cycle (ISORC) for Waste Heat Recovery in Distributed Generation and Combined Heat and Power Production, Paper No: IHTC14-22704, pp. 77–81. Proc. of IHTC‑14, Washington, 2010. DOI: 10.1115/IHTC14–22704. Доступ 19.10.2020.Chudnovsky, Y., Gotovsky, M., Greenman, M. [et al]. Integrated Steam/Organic Rankine Cycle (ISORC) for Waste Heat Recovery in Distributed Generation and Combined Heat and Power Production, Paper No: IHTC14-22704, pp. 77-81. Proc. of IHTC-14, Washington, 2010. DOI: 10.1115/IHTC14-22704. Last accessed 19.10.2020.Hinze, J. O. Turbulence, 2nd ed., New York, McGraw-Hill, 1975, 790 p.Hinze, J. O. Turbulence, 2nd ed., New York, McGraw-Hill, 1975, 790 p.Schlichting, H. Boundary-Layer Theory. 6th ed., New York, McGraw-Hill, 1979, 838 p.Schlichting, H. Boundary-Layer Theory. 6th ed., New York, McGraw-Hill, 1979, 838 p.Dmitrenko, A. V. Equivalence of measures and stochastic equations for turbulent flows. Doklady Physics, 2013, Vol. 58, Iss. 6, pp. 228–235. DOI: 10.1134/S1028335813060098. Доступ 19.10.2020.Dmitrenko, A. V. Equivalence of measures and stochastic equations for turbulent flows. Doklady Physics, 2013, Vol. 58, Iss. 6, pp. 228–235. DOI: 10.1134/S1028335813060098. Last accessed 19.10.2020.Dmitrenko, A. V. Calculation of pressure pulsations for a turbulent heterogeneous medium. Doklady Physics, 2007, Vol. 52, Iss. 7, pp. 384–387. DOI: 10.1134/S1028335807120166. Доступ 19.10.2020.Dmitrenko, A. V. Calculation of pressure pulsations for a turbulent heterogeneous medium. Doklady Physics, 2007, Vol. 52, Iss. 7, pp. 384–387. DOI: 10.1134/S1028335807120166. Last accessed 19.10.2020.Dmitrenko, A. V. Some analytical results of the theory of equivalence measures and stochastic theory of turbulence for non-isothermal flows. Advanced Studies in Theoretical Physics, 2014, Vol. 8, Iss. 25, pp. 1101–1111. DOI: 10.12988/astp.2014.49131. Доступ 19.10.2020.Dmitrenko, A. V. Some analytical results of the theory of equivalence measures and stochastic theory of turbulence for non-isothermal flows. Advanced Studies in Theoretical Physics, 2014, Vol. 8, Iss. 25, pp. 1101–1111. DOI: 10.12988/astp.2014.49131. Last accessed 19.10.2020.Dmitrenko, A. V. Analytical estimation of velocity and temperature fields in a circular pipe on the basis of stochastic equations and equivalence of measures. J. Eng. Phys. Thermophys, 2015, Vol. 88, Iss. 6, pp. 1569–1576. DOI: 10.1007/s10891-015-1344‑x. Доступ 19.10.2020.Dmitrenko, A. V. Analytical estimation of velocity and temperature fields in a circular pipe on the basis of stochastic equations and equivalence of measures. J. Eng. Phys. Thermophys. 2015, Vol. 88, Iss. 6, pp. 1569–1576. DOI: 10.1007/s10891-015-1344-x. Last accessed 19.10.2020.Dmitrenko, A. V. Determination of critical Reynolds numbers for non-isothermal flows with using stochastic theories of turbulence and equivalent measures. Heat Transfer Research, 2015, Vol. 47, Iss. 1, pp. 338–399. DOI: 10.1615/HeatTransRes.2015014191. Доступ 19.10.2020.Dmitrenko, A. V. Determination of critical Reynolds numbers for non-isothermal flows with using stochastic theories of turbulence and equivalent measures. Heat Transfer Research, 2015, Vol. 47, Iss.1, pp. 338–399. DOI: 10.1615/HeatTransRes.2015014191. Last accessed 19.10.2020.Dmitrenko, A. V. An estimation of turbulent vector fields, spectral and correlation functions depending on initial turbulence based on stochastic equations. The Landau fractal equation. Int J Fluid Mech Res., 2016, Vol. 43, Iss. 3, pp. 82–91. DOI: 10.1615/InterJFluidMechRes.v43.i3.60. Доступ 19.10.2020.Dmitrenko, A. V. An estimation of turbulent vector fields, spectral and correlation functions depending on initial turbulence based on stochastic equations. The Landau fractal equation. Int. J. Fluid. Mech. Res., 2016, Vol. 43, Iss. 3, pp. 82–91. DOI: 10.1615/InterJFluidMechRes.v43.i3.60. Last accessed 19.10.2020.Dmitrenko, A. V. The theory of equivalence measures and stochastic theory of turbulence for non-isothermal flow on the flat plate. Int J Fluid Mech Res., 2016, Vol. 43, Iss. 2, pp. 182–187. DOI: 10.1615/InterJFluidMechRes.v43.i2.60. Доступ 19.10.2020.Dmitrenko, A. V. The theory of equivalence measures and stochastic theory of turbulence for non-isothermal flow on the flat plate. Int. J. Fluid. Mech. Res., 2016, Vol. 43, Iss. 2, pp. 182–187. DOI: 10.1615/InterJFluidMechRes.v43.i2.60. Last accessed 19.10.2020.Dmitrenko, A. V. Stochastic equations for continuum and determination of hydraulic drag coefficients for smooth flat plate and smooth round tube with taking into account intensity and scale of turbulent flow. Continuum Mechanics and Thermodynamics, Vol. 29, Iss. 1. pp. 1–9. DOI: 10.1007/s00161-016-0514-1. Доступ 19.10.2020.Dmitrenko, A. V. Stochastic equations for continuum and determination of hydraulic drag coefficients for smooth flat plate and smooth round tube with taking into account intensity and scale of turbulent flow. Continuum Mechanics and Thermodynamics, Vol. 29, Iss. 1. pp. 1–9. DOI: 10.1007/s00161-016-0514-1. Last accessed 19.10.2020.Dmitrenko, A. V. Analytical determination of the heat transfer coefficient for gas, liquid and liquid metal flows in the tube based on stochastic equations and equivalence of measures for continuum. Continuum Mechanics and Thermodynamics, 2017, Vol. 29, Iss. 6, pp. 1197–1206. DOI: 10.1007/s00161-017-0566‑x. Доступ 19.10.2020.Dmitrenko, A. V. Analytical determination of the heat transfer coefficient for gas, liquid and liquid metal flows in the tube based on stochastic equations and equivalence of measures for continuum. Continuum Mechanics and Thermodynamics, 2017, Vol. 29, Iss. 6, pp. 1197–1206. DOI: 10.1007/s00161-017-0566-x. Last accessed 19.10.2020.Dmitrenko, A. V. Estimation of the critical Rayleigh number as a function of an initial turbulence in the boundary layer of the vertical heated plate. Heat Transfer Research, 2017, Vol. 48, Iss. 13, pp. 1195–1202. DOI: 10.1615/HeatTransRes.2017018750. Доступ 19.10.2020.Dmitrenko, A. V. Estimation of the critical Rayleigh number as a function of an initial turbulence in the boundary layer of the vertical heated plate. Heat Transfer Research, 2017, Vol. 48, Iss. 13, pp. 1195–1202. DOI: 10.1615/HeatTransRes.2017018750. Last accessed 19.10.2020.Dmitrenko, A. V. Determination of the Coefficients of Heat Transfer and Friction in Supercritical-Pressure Nuclear Reactors with Account of the Intensity and Scale of Flow Turbulence on the Basis of the Theory of Stochastic Equations and Equivalence of Measures. J. Eng. Phys. Thermophys, 2017, Vol. 90, Iss. 4, pp. 1288–1294. DOI: 10.1007/s10891-017-1685-8. Доступ 19.10.2020.Dmitrenko, A. V. Determination of the Coefficients of Heat Transfer and Friction in Supercritical- Pressure Nuclear Reactors with Account of the Intensity and Scale of Flow Turbulence on the Basis of the Theory of Stochastic Equations and Equivalence of Measures. J. Eng. Phys. Thermophys, 2017, Vol. 90, Iss. 4, pp. 1288–1294. DOI: 10.1007/s10891-017-1685-8. Last accessed 19.10.2020.Dmitrenko, A. V. Regular Coupling between Deterministic (Laminar) and Random (Turbulent) Motions-Equivalence of Measures. Scientific Discovery 2013, Diploma No. 458, registration No. 583 of December 2.Dmitrenko, A. V. Regular Coupling between Deterministic (Laminar) and Random (Turbulent) Motions- Equivalence of Measures Scientific Discovery 2013, Diploma No. 458, registration No. 583 of December 2.Дмитренко А. В. Теория эквивалентных мер и множеств с повторяющимися, счётными фрактальными элементами. Стохастическая термодинамика и турбулентность. Коррелятор «Детерминированность–случайность»: Монография. – М.: Галлея-Принт, 2013. – 226 с.Dmitrenko, A. V. Theory of equivalent measures and sets with repeating, countable fractal elements. Stochastic thermodynamics and turbulence. Correlator «Determination – randomness»: Monograph [Teoriya ekvivalentnykh mer i mnozhestv s povtoryayushchimisya, schetnymi fraktalnymi elementami. Stokhasticheskaya termodinamika i turbulentnost’. Korrelyator «Determinirovannost’–sluchainost’»: Monografiya]. Moscow, Galleya-Print publ., 2013, 226 p.Dmitrenko, A. V. Fundamentals of heat and mass transfer and hydrodynamics of single-phase and two-phase media. Criterial integral statistical methods and direct numerical simulation. Moscow, Galleya print, 2008, 398 p.Dmitrenko, A. V. Fundamentals of heat and mass transfer and hydrodynamics of single-phase and two-phase media. Criterial integral statistical methods and direct numerical simulation. Moscow, Galleya print, 2008, 398 p.Dmitrenko, A. V. Uncertainty relation in turbulent shear flow based on stochastic equations of the continuum and the equivalence of measures. Continuum Mechanics and Thermodynamics, 2019. DOI: 10.1007/s00161-019-00784-0. Доступ 19.10.2020.Dmitrenko, A. V. Uncertainty relation in turbulent shear flow based on stochastic equations of the continuum and the equivalence of measures. Continuum Mechanics and Thermodynamics, 2019. DOI: 10.1007/s00161-019-00784-0. Last accessed 19.10.2020.Dmitrenko, A. V. Formation of a turbulence spectrum in the inertial interval on the basis of the theory of stochastic equations and equivalence of measures. J. Eng. Phys. Thermophys, 2020, Vol. 93, Iss. 5, pp. 122–127. DOI: 10.1007/s10891-020-02098-4. Доступ 19.10.2020.Dmitrenko, A. V. Formation of a turbulence spectrum in the inertial interval on the basis of the theory of stochastic equations and equivalence of measures. J. Eng. Phys. Thermophys, 2020, Vol. 93, Iss. 5, pp. 122–127. DOI: 10.1007/s10891-020-02098-4. Last accessed 19.10.2020.Dmitrenko, A.V. The correlation dimension of an attarctor determined on the base of the theory of equivalence of measures and stochastic equations for continuum. Continuum Mechanics and Thermodynamics, Vol. 32, Iss. 2, pp. 63–74. DOI: 10.1007/s00161-019-00784-0. Доступ 19.10.2020.Dmitrenko, A.V. The correlation dimension of an attarctor determined on the base of the theory of equivalence of measures and stochastic equations for continuum. Continuum Mechanics and Thermodynamics. Vol. 32, Iss. 2, pp. 63–74. DOI: 10.1007/s00161-019-00784-0. Last accessed 19.10.2020.Дмитренко А. В. Современные аспекты стохастической теории гидродинамики // XII Всероссийский съезд по фундаментальным проблемам теоретической и прикладной механики, Уфа, 20–24 августа 2019. – Сб. трудов. – Т. 2. – С. 339–341.Dmitrenko, A. V. Modern aspects of the stochastic theory of hydrodynamics [Sovremennie aspekty stokhasticheskoi teorii gidrodinamiki]. 12th All-Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics, Ufa, August 20–24, 2019. Collection of works, Vol. 2, pp. 339–341.Дмитренко А. В., Колосова М. А. Определение гидравлических характеристик промышленных агрегатов на основе стохастической теории гидродинамики // XII Всероссийский съезд по фундаментальным проблемам теоретической и прикладной механики, Уфа, 20–24 августа 2019. – Сб. трудов. – Т. 2. – С. 342–343.Dmitrenko, A. V., Kolosova, M. A. Determination of the hydraulic characteristics of industrial units based on the stochastic theory of hydrodynamics [Opredelenie gidravlicheskikh kharakteristik promyshlennykh agregatov na osnove stokhasticheskoi teorii gidrodinamiki]. 12th All-Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics, Ufa, August 20–24, 2019. Collection of works, Vol. 2, pp. 342–343.Dmitrenko, A. V., Kolosova, M. A. The possibility of using low-potential heat based on the organic Rankine cycle and determination of hydraulic characteristics of industrial units based on the theory of stochastic equations and equivalence of measures. JP Journal of Heat and Mass Transfer, 2020, Vol. 21, Iss. 1, pp. 11–18. DOI: http://dx.doi.org/ 10.17654/HM021010200. Доступ 19.10.2020.Dmitrenko, A. V., Kolosova, M. A. The possibility of using low-potential heat based on the organic Rankine cycle and determination of hydraulic characteristics of industrial units based on the theory of stochastic equations and equivalence of measures. J.P. Journal of Heat and Mass Transfer, 2020, Vol. 21, Iss. 1, pp. 11–18. DOI: http://dx.doi.org/ 10.17654/HM021010200. Last accessed 19.10.2020.Dmitrenko, A. V. Theoretical solutions for spectral function of the turbulent medium based on the stochastic equations and equivalence of measures. Continuum Mechanics and Thermodynamics, 2020. DOI: https://doi.org/10.1007/s00161-020-00890-4. Доступ 19.10.2020.Dmitrenko, A. V. Theoretical solutions for spectral function of the turbulent medium based on the stochastic equations and equivalence of measures. Continuum Mechanics and Thermodynamics, 2020. DOI: https://doi.org/10.1007/s00161-020-00890-4. Last accessed 19.10.2020.Данилова Г. Н., Богданов С. Н., Иванов О. П., Медникова Н. М., Крамской Э. И. Теплообменные аппараты холодильных установок // Под ред. д. т.н. Г. Н. Даниловой. – 2‑е изд., перераб. и доп. – Л.: Машиностроение. Ленинградское отделение, 1986. – 303 с.Danilova, G. N., Bogdanov, S. N., Ivanov, O. P., Mednikova, N. M., Kramskoy, E. I. Heat exchangers of refrigeration units [Teploobmennie apparaty kholodilnykh ustanovok]. Ed. by D.Sc. (Eng) G. N. Danilova. 2nd ed., rev. and enl. Leningrad, Mashinostroenie publ., Leningrad branch, 1986, 303 p.Кириллов П. Л., Юрьев Ю. С., Бобков В. П. Справочник по теплогидравлическим расчётам (ядерные реакторы, теплообменники, парогенераторы). – М.: Энергоатомиздат, 1990. – 360 с.Kirillov, P. L., Yuryev, Yu. S., Bobkov, V. P. Handbook of thermohydraulic calculations (nuclear reactors, heat exchangers, steam generators) [Spravochnik po teplogidravlicheskim raschetam (yadernie reaktory, teploobmenniki, parogeneratory)]. Moscow, Energoatomizdat publ., 1990, 360 p.

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