danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2020-64-4-495-505dan-907Research ArticleТЕХНИЧЕСКИЕ НАУКИTECHNICAL SCIENCESНовый подход в приближенном решении задачи Стефана с конвективным граничным условием (Представлено членом-корреспондентом Н.В. Павлюкевичем)A new approach to approximate solution of the Stefan problem with a convective boundary condition (Communicated by Corresponding Member Nikolai V. Pavlyukevich)КотВ. А.KotV. A.

Кот Валерий Андреевич - кандидат технических наук, старший научный сотрудник, ИТМО им. А.В. Лыкова НАН Беларуси.

ул. П. Бровки, 15, 220072, Минск.

Kot Valery Andreevich - Ph. D. (Engineering), Senior researcher, A.V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus.

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 Belarus202030082020644495505Copyright © Кот В.А., 20202020Кот 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/907

Предложено два новых варианта приближенного аналитического решения однофазной задачи Стефана с конвективным граничным условием на фиксированной границе. Данные решения основаны на применении новых интегральных соотношений, вытекающих из постановочной части задачи и образующих бесконечную последовательность. Показано, что наиболее точным вариантом решения задачи Стефана с конвективным граничным условием является отказ от точного выполнения классического условия Стефана на свободной границе с его заменой на одно из интегральных соотношений. На примере рассмотрения тестовой задачи Стефана с граничным условием Робина, имеющей точное аналитическое решение, показано, что предложенный новый подход в решении задачи является существенно более точным и эффективным по сравнению с известными вариантами интегральной расчетной схемы, в том числе по сравнению с методом интеграла теплового баланса при точном выполнении условия Стефана на свободной границе. В работе представлены решения задачи на основе применения квадратичного и кубического полиномов. В решениях тестовой задачи на основе кубического полинома относительная ошибка определения положения свободной границы составляет тысячные и сотые доли процента. При этом в момент времени t = 1 относительная ошибка для температурного профиля составляет всего εT = 0,075 %.

Two new variants of approximate analytical solution of the one-phase Stefan problem with a convective boundary condition at a fixed boundary are proposed. These approaches are based on the use of new integral relations forming infinite sequences. It is shown that the most exact variant of solving the Stefan problem with a convective boundary condition is to refuse from the classical Stefan condition at the free boundary and to replace it with its integral relation. By the example of solving the test Stefan problem with a Robin boundary condition, having an exact analytical solution, it is shown that the proposed approach is much more exact and efficient compared to the known variants of the integral computational scheme, including the heat-balance integral method allowing the Stefan condition at the free boundary to be satisfied. The solutions obtained with the use of the square-law and cubic polynomials are presented. As for the test problem using the cubic polynomial, the relative error in determining the free boundary comprises hundredths and thousandths of percent. In this case, at the time instant t = 1, the relative error in determining the temperature profile is εT = 0.075 %.

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