INTEGRAL METHOD OF SOLVING HEAT-CONDUCTION PROBLEMS WITH THE SECOND-KIND BOUNDARY CONDITION. 1. BASIC STATEMENTS

On the basis of systems of identical equalities and integral boundary characteristics, a new algorithm of solving a boundary-value problem on the nonstationary heat conduction in a canonical body with boundary condition of the second kind has been developed. The scheme proposed for finding approximate analytical solutions of boundary-value problems on nonstationary heat conduction with boundary conditions of the second kind involves the introduction into consideration of a temperature-disturbance front and separation of the whole heating process into two stages. For the first stage of this process, on the basis of the differentiation of the heat-conduction equation over a space variable and the application of symmetric integral and differential operators to the expressions obtained, two sequences of integral and differential identical equalities have been constructed. Each of these sequences includes integral or differential limiting characteristics for a definite boundary condition of the second kind. For the second stage, by way of introduction of a boundary function, differentiation of the heat-conduction equation with respect to a spatial coordinate, and application of integral operators to the expression obtained, a sequence of integral identical equalities involving integral boundary characteristics for the second-kind boundary condition has been constructed. On the basis of the integral and differential identical equalities obtained, closed systems of equations, allowing one to find polynomial coefficients of the temperature profile for the first and second stages of the heating process, have been constructed. A general scheme of determining approximate eigenvalues of boundary-value problems with boundary conditions of the second kind on the basis of construction of an ordinary differential equation and transformation of it into the characteristic equation is proposed. For each of the two stages of the heating process, special integral operators, reducing the boundary-value heat-conduction problem to the ordinary differential equation, are proposed.

heat-conduction equation, integral method of heat balance, temperature disturbance front

1. Goodman 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)70097-2

2. Goodman T. R. Heat-Balance Integral – Further Considerations and Refinements. Transactions of the ASME, Journal of Heat Transfer. 1961, vol. 83, is. 1, pp. 83–85. https://doi.org/10.1115/1.3680474

3. Wood A. S. A new look at the heat balance integral method. Applied Mathematical Modelling, 2001, vol. 25, is. 10, pp. 815–824. https://doi.org/10.1016/S0307-904X(01)00016-6

4. Bio M. Variational principles in heat-exchange theory. Moscow, Energiya Publ., 1975. 209 p. (in Russian).

5. Dorodnitsyn A. A. General method of integral relations and its application to boundary layer theory. Kármán T. Von, Ballantyne A. M., Dexter R. R. (eds.). Advances in Aeronautical Sciences: Proceedings of the Second International Congress in the Aeronautical Sciences, Zürich, 12–16 September 1960. Volume 3. New York, Pergamon, 1962, pp. 207–219. https://doi.org/10.1016/B978-0-08-006550-2.50018-1

6. Hristov J. The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and exercises. Thermal Science, 2009, vol. 13, no. 2, pp. 27–48. https://doi.org/2298/TSCI0902027H

7. Sadoun, N., El-Khider Si-Ahmed, Colinet P. On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions. Applied Mathematical Modelling, 2006, vol. 30, iss. 6, pp. 531–544. https://doi.org/10.1016/j.apm.2005.06.003

8. Mitchell S. L., Myers T. G. Application of standard and refined heat balance integral methods to one-dimensional Stefan problems. SIAM Review, 2010, vol. 52, iss. 1, pp. 57–86. https://doi.org/10.1137/080733036

9. Myers T. G. Optimizing the exponent in the heat balance and refined integral methods. International Communications in Heat and Mass Transfer, 2009, vol. 36, iss. 2, pp. 143–147. https://doi.org/10.1016/j.icheatmasstransfer.2008.10.013

10. Langford D. The heat balance integral method. International Journal of Heat and Mass Transfer, 1973, vol. 16, iss. 12, pp. 2424–2428. https://doi.org/10.1016/0017-9310(73)90026-4

11. Mitchell 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, iss 17–18, pp. 3540–3551. https://doi.org/10.1016/j.ijheatmasstransfer.2010.04.015

12. Fedorov F. M. Boundary method of solving applied problems of mathematical physics. Novosibirsk, Nauka Publ., 2000. 220 p. (in Russian).

13. Stefanyuk E. V., Kudinov V. A. Additional boundary conditions in nonstationary problems of heat conduction. High Themperature, 2009. vol. 47, iss. 2, pp. 250–262. https://doi.org/10.1134/s0018151x09020163

14. 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/s10891-016-1377-9

15. Kot V. A. Boundary Characteristics for the Generalized Heat-Conduction Equation and Their Equivalent Representations. Journal of Engineering Physics and Thermophysics, 2016, vol. 89, no. 4, pp. 985–1007. https://doi.org/10.1007/s10891-016-1461-1

16. Kot V. A. Integral Method of Boundary Characteristics: The Dirichlet Condition. Principles. Heat Transfer Research, 2016, vol. 47, no. 10, pp. 927–944. https://doi.org/10.1615/HeatTransRes.2016014883

17. Kot V. A. The Boundary Function Method. Fundamentals. Journal of Engineering Physics and Thermophysics, 2017, vol. 90, no. 2, pp. 366–391. https://doi.org/10.1007/s10891-017-1576-z