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Home / 2019 issue 6 / ANALYTICAL METHOD FOR SOLVING PROBLEMS OF HEAT CONDUCTIVITY WITH NEWTON’S BOUNDARY CONDITIONS

ANALYTICAL METHOD FOR SOLVING PROBLEMS OF HEAT CONDUCTIVITY WITH NEWTON’S BOUNDARY CONDITIONS

A.V. Eremin, K.V. Gubareva

Vestnik IGEU, 2019 issue 6

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Abstract in English: 

Background: Non-stationary heat transmission within solid bodies is described using parabolic and hyperbolic equations. Currently, numerical methods for studying the processes of heat and mass transfer in the flows of liquids and gases have gained distribution. Modern programs allow the automatic construction of computational grids, solve systems of equations, and offer a wide range of tools for analysis. Approximate analytical solutions have significant advantages compared to numerical ones. In particular, the solutions obtained in an analytical form allow one to perform parametric analysis of the system under study, configure and program measurement devices, etc.

Materials and methods: Based on the joint use of the additional desired function and additional boundary conditions in the integral method of heat balance, a method for mathematical modeling of the heat transfer process in a plate under symmetric boundary conditions of the third kind is developed.

Results: Using the heat flux density as a new sought function, the problem of finding a solution to the partial differential equation with respect to the temperature function reduces to integrating an ordinary differential equation with respect to the heat flux density on the surface of the studied region. It is shown that isotherms appear on the surface of the plate, having a certain initial velocity, which depends on the heat transfer intensity.

Conclusions: The presented method can be used in determining the heat flux density of buildings and heating devices, finding heat losses during convective heat transfer, and designing heat transfer equipment. The results can be applied to increase the validity and reliability of the calculation of actual losses and balance of thermal energy. The calculation results are compared with the exact solution, the reliability, validity of the method and a high degree of approximation with an error of about 3% are shown. The accuracy of the solution depends on the number of approximations performed and is determined by the degree of the approximating polynomial.

References in English: 

1. Lykov, A.V. Teoriya teploprovodnosti [Theory of heat conductivity]. Moscow: Vysshaya shkola, 1967. 600 p.

2. Tsoy, P.V. Sistemnye metody rascheta kraevykh zadach teplomassoperenosa [System methods of calculation of boundary value problems of heat and mass transfer]. Moscow: Izdatel'stvo MEI, 2005. 567 p.

3. Belyaev N.M., Ryadno A.A. Metody teorii teploprovodnosti. T. 1 [Thermal conductivity. methods theory. Vol. 1]. Moscow: Vysshaya shkola, 1982. 328 p.

4. Kudinov, V.A., Kartashov, E.M., Kalashnikov, V.V. Analiticheskie resheniya zadach teplomassoperenosa I termouprugosti dlya mnogosloynykh konstruktsiy [Analytical solutions to problems of heat and mass transfer and thermoelasticity for multilayer atructures]. Moscow: Vysshaya shkola, 2005. 430 p.

5. Kudinov V.A., Averin B.V., Stefanyuk E.V. Teploprovodnost' i termouprugost' v mnogosloynykh konstruktsiyakh [Heat conductivity and thermoelasticity in multilayered structures]. Samara: SamGTU, 2008. 391 p.

6. Kantorovich, L.V., Krylov, V.I. Priblizhennye metody vysshego analiza [Approximate methods of higher analysis]. Leningrad: Fizmatgiz, 1962. 708 p.

7. Kantorovich, L.V. Ob odnom metode priblizhennogo resheniya differentsial'nykh uravneniy v chastnykh proizvodnykh [On one method of approximate solution to partial differential equations]. DAN SSSR, 1934, vol. 2, 2, pp. 532–534.

8. Fedorov, F.M. Granichnyy metod resheniya prikladnykh zadach matematicheskoy fiziki [A boundary method of solving applied problems of mathematical physics]. Novosibirsk: Nauka, 2000. 220 p.

9. Kotova E.V., Eremin A.V., Kudinov V.A., Tkachev V.K., Kuznetsova A.E. Metod dopolnitel'nykh iskomykh funktsiy v zadachakh teploprovodnosti s peremennymi fizicheskimi svoystvami sredy [Method of additional unknown functions in heat conductivity problems with variable physical properties of the medium]  in Vestnik IGEU, 2019, issue 2, pp. 59 – 70.

10. Eremin A.V., Stefanyuk E.V., Abisheva L.S. Identifikatsiya istochnika teploty na osnove analiticheskogo resheniya zadachi teploprovodnosti  [Heat source identification based on analytical solutions of the heat-conduction problem] in Izvestiya. Ferrous Metallurgy, 2016, vol. 59, 5, pp. 339 – 346.

11. Kudinov I.V., Kudinov V.A., Kotova E.V., Eremin A.V. Ob odnom metode resheniya nestatsionarnykh kraevykh zadach [On one method of solving nonstationary boundary-value problems] in Inzhenerno – fizicheskiy zhurnal, 2017, vol. 90, 6, pp. 1387 – 1397.

12. O.P. Layeni, J.V. Johnson. Hybrids of the heat balance integral method in Applied Mathematics and Computation, 2012, vol. 218, 14, pp. 7431–7444.

13. S.L. Mitchell, T.G. Myers. Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions in International Journal of Heat and Mass Transfer, 2010, vol. 53, 17–18, pp. 3540–3551.

14. S.L. Mitchell, T.G. Myers. Application of Standard and Refined Heat Balance Integral Methods to One–Dimensional Stefan Problems, in Siam Review, 2010, vol. 52, 1, pp. 57–86.

15. V. Novozhilov. Application of heat–balance integral method to conjugate thermal explosion in Thermal Science, 2009, vol. 13, 2, pp. 73–80.

Key words in Russian: 
дополнительная искомая функция, дополнительные граничные условия, интеграл теплового баланса, изотермы, скорость перемещения изотерм.
Key words in English: 
approximate analytical solution, additional boundary conditions, the integral of heat balance, isotherms, isotherms movement velocity.
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