danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2019-63-4-485-495dan-636Research ArticleТЕХНИЧЕСКИЕ НАУКИTECHNICAL SCIENCESИнтегральные методы решения задач тепломассопереноса: новая концепция (условие Дирихле)Integral methods of solving heat-conduction problems: a new concept (Dirichlet condition)КотВ. А.KotValery A.

Кот Валерий Андреевич – канд. техн. наук, ст. науч. cотрудник

ул. П. Бровки, 15, 220072, Минск

Kot Valery Andreevich – Ph. D. (Engineering), Senior researcher

15, P. Brovka Str., 220072, Minsk

valery.kot@hmti.ac.byИнститут тепло- и массообмена имени А. В. Лыкова Национальной академии наук БеларусиA. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus201913092019634485495Copyright © Кот В.А., 20192019Кот В.А.Kot V.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/636

На основе рассмотрения задачи теплопроводности для полуограниченного пространства с описанием температурного профиля параболой степени n предложена новая концепция построения определяющих интегральных соотношений. Данная концепция основана на введении локальной функции для теплового потока либо температуры, которые определяются из уравнения теплопроводности. Такой подход позволил получить ряд новых интегральных соотношений: улучшенное интегральное соотношение для температурного момента, интеграл квадратичного теплового потока, интеграл квадратичной температурной функции. Предложены две схемы оптимизации степени n на основе новых норм ошибки E1 и E1L. По сравнению с нормой Лангфорда удалось значительно повысить аппроксимационную точность решений поставленной задачи.

On the basic of consideration of the heat-conduction problem for a semi-bounded space with a temperature profile defined by a parabola with an exponent n, a new concept of construction of constitutive involves the introduction of a local function for a heat flow or for the temperature, with is determined from the heat-conduction equation. The approach proposed made it possible to obtain a number of new integral relation: an improved integral for the temperature momentum, an integral of a quadratic heat flow, and an integral of a quadratic temperature function. Two Schemes of optimizing the exponent n with the use of the error norms E1 and are proposed. As compared to the Langford norm, the indicated error norms made it possible to substantially increase the approximation accuracy of solutions of the problem posed.

интегральный метод теплового балансаинтеграл квадратичной температурной функцииинтеграл квадратичного теплового потокагибридный интегральный методheat-balance integral methodintegral of a quadratic temperature functionintegral of a quadratic heat flowhybrid integral methodReferencesGoodman, T. R. Application of integral methods to transient nonlinear heat transfer / Т. R. Goodman // Adv. Heat Transfer. – 1964. – Vol. 1. – P. 51–122. https://doi.org/10.1016/s0065-2717(08)70097-2Goodman T. R. Application of integral methods to transient nonlinear heat transfer. Advances in Heat Transfer, 1964, vol. 1, pp. 51–122. https://doi.org/10.1016/s0065-2717(08)70097-2Wood, A. S. A new look at the heat balance integral method / A. S. Wood // Applied Mathematical Modelling. – 2001. – Vol. 25, N 10. – P. 815–824. https://doi.org/10.1016/s0307-904x(01)00016-6Wood A. S. A new look at the heat balance integral method. Applied Mathematical Modelling, 2001, vol. 25, no. 10, pp. 815–824. https://doi.org/10.1016/s0307-904x(01)00016-6Braga, W. Analytical Solution for One-Dimensional Semi-Infnite Heat Transfer Problem with Convection Boundary Condition / W. Braga, M. Mantelli, J. Azevedo // 38th AIAA Thermophysics Conference, Canada. – 2005. – Paper AIAA-2005-4686. https://doi.org/10.2514/6.2005-4686Braga W., Mantelli M., Azevedo J. Analytical Solution for One-Dimensional Semi-Infnite Heat Transfer Problem with Convection Boundary Condition. 38th AIAA Thermophysics Conference, Canada, 2005, paper AIAA-2005-4686. https://doi.org/10.2514/6.2005-4686Langford, D. The heat balance integral method / D. Langford // International Journal of Heat and Mass Transfer. – 1973. – Vol. 16, N 12. – P. 2424–2428. https://doi.org/10.1016/0017-9310(73)90026-4Langford D. The heat balance integral method. International Journal of Heat and Mass Transfer. 1973, vol. 16, no. 12, pp. 2424–2428. https://doi.org/10.1016/0017-9310(73)90026-4Кот, В. А. Метод граничных характеристик / В. А. Кот // Инженерно-физический журнал. – 2015. – Т. 88, № 6. – C. 1345–1363.kot V. A. Method of boundary characteristics. Journal of Engineering Physics and Thermophysics, 2015, vol. 88, no. 6, pp. 1390–1408. https://doi.org/10.1007/s10891-015-1324-1Carslow, H. S. Conduction of Heat in Solids / H. S. Carslow, J. C. Jaeger. – 2nd ed. – Oxford, 1992.Carslow H. S., Jaeger J. C. Conduction of Heat in Solids, 2nd ed. Oxford, 1992.Hristov, J. The heat-balance 2nd ed., integral method by a parabolic profle with unspecifed exponent: Analysis and exercises / J. Hristov // Thermal Science. – 2009. – Vol. 13, N 2. – P. 27–48. https://doi.org/10.2298/tsci0902027hHristov J. The heat-balance integral method by a parabolic profle with unspecifed exponent: Analysis and exercises. Thermal Science, 2009, vol. 13, no. 2, pp. 27–48. https://doi.org/10.2298/tsci0902027hMyers, T. G. Optimizing the exponent in the heat balance and refned integral methods / T. G. Myers // International Communication in Heat and Mass Transfer. – 2009. – Vol. 36, N 2. – P. 143–147. https://doi.org/10.1016/j.icheatmasstransfer.2008.10.013Myers T. G. Optimizing the exponent in the heat balance and refned integral methods. International Communication in Heat and Mass Transfer, 2009, vol. 36, no. 2, pp. 143–147. https://doi.org/10.1016/j.icheatmasstransfer.2008.10.013Volkov, V. N. A refnement of the integral method in solving the heat conduction equation / V. N. Volkov, V. k. Li-Orlov // Heat Transfer Soviet Research. – 1970. – Vol. 2, N 2. – P. 41–47.Volkov V. N., Li-Orlov V. k. A refnement of the integral method in solving the heat conduction equation. Heat Transfer Soviet Research, 1970, vol. 2, no. 2, pp. 41–47.Sadoun, N. On the refned integral method for the one-phase Stefan problem with time-dependent boundary conditions / N. Sadoun, E. k. Si-Ahmed, P. Colinet // Applied Mathematical Modelling. – 2006. – Vol. 30, N 6. – P. 531–544. https://doi.org/10.1016/j.apm.2005.06.003Sadoun N., Si-Ahmed E. k., Colinet P. On the refned integral method for the one-phase Stefan problem with time-dependent boundary conditions. Applied Mathematical Modelling, 2006, vol. 30, no. 6, pp. 531–544. https://doi.org/10.1016/j.apm.2005.06.003Mitchell, S. L. Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions / S. L. Mitchell, T. G. Myers // International Journal of Heat and Mass Transfer. – 2010. – Vol. 53, N 17–18. – P. 3540–3551. https://doi.org/10.1016/j.ijheatmasstransfer.2010.04.015Mitchell S. L., Myers T. G. Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions. International Journal of Heat and Mass Transfer, 2010, vol. 53, no. 17–18, pp. 3540–3551. https://doi.org/10.1016/j.ijheatmasstransfer.2010.04.015Gupta, R. S. Constrained integral method for solving moving boundary problems / R. S. Gupta, N. C. Banik // Computer Methods in Applied Mechanics and Engineering. – 1988. – Vol. 67, N 2. – P. 211–221. https://doi.org/10.1016/0045-7825(88)90126-0Gupta R. S., Banik N. C. Constrained integral method for solving moving boundary problems. Computer Methods in Applied Mechanics and Engineering, 1988, vol. 67, no. 2, pp. 211–221. https://doi.org/10.1016/0045-7825(88)90126-0Gupta, R. S. Diffusion of oxygen in a sphere with simultaneous absorption / R. S. Gupta, N. C. Banik // Applied Mathematical Modelling. – 1990. – Vol. 14, N 3. – P. 114–121. https://doi.org/10.1016/0307-904x(90)90044-6Gupta R. S., Banik N. C. Diffusion of oxygen in a sphere with simultaneous absorption. Applied Mathematical Modelling, 1990, vol. 14, no. 3, pp. 114–121. https://doi.org/10.1016/0307-904x(90)90044-6Hristov, J. The heat-balance integral: 1. How to calibrate the parabolic profle? / J. Hristov // Comptes Rendus Mecanique. – 2012. – Vol. 340, N 7. – P. 485–492. https://doi.org/10.1016/j.crme.2012.03.001Hristov J. The heat-balance integral: 1. How to calibrate the parabolic profle? Comptes Rendus Mecanique, 2012, vol. 340, no. 7, pp. 485–492. https://doi.org/10.1016/j.crme.2012.03.001Zien, T. F. Approximate calculation of transient heat conduction / T. F. Zien // AIAA Journal. – 1976. – Vol. 14, N 3. – P. 404–406. https://doi.org/10.2514/3.7111Zien T. F. Approximate calculation of transient heat conduction. AIAA Journal, 1976, vol. 14, no. 3, pp. 404–406. https://doi.org/10.2514/3.7111Zien, T. F. Integral solution of ablation problems with time-dependent heat flux / T. F. Zien // AIAA Journal. – 1978. – Vol. 16, N 12. – P. 1287–1295. https://doi.org/10.2514/3.61045Zien T. F. Integral solution of ablation problems with time-dependent heat flux. AIAA Journal, 1978, vol. 16, no. 12, pp. 1287–1295. https://doi.org/10.2514/3.61045

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