Solution of a nonlinear heat transfer problem based on experimental data
DOI:
https://doi.org/10.26577/JMMCS.2022.v114.i2.014Keywords:
жылу өткiзгiштiк, сызықтық емес, айырықша есеп, жинақтылық, керi есеп, параметрге қатысты дифференциалдауAbstract
The paper develops a method for solving nonlinear equations of heat conduction. Two-layer container complexes have been created, the side faces of which are thermally insulated so that the 1D heat equation can be used. In order not to solve the boundary value problem with a contact discontinuity and lose the accuracy of the solution method, a temperature sensor was placed at the junction of two media, and a mixed boundary value problem is solved in each area (container). To provide initial data for the initial boundary value problem, three temperature sensors were used: two sensors measure air temperatures at the left and right boundaries of the container complex; the third sensor measures the temperature of the soil at the junction of two media. The paper numerically investigates the initial-boundary problem of heat conduction with nonlinear coefficients of heat conduction, heat capacity, heat transfer and density of the material. To solve a nonlinear initial-boundary value problem, the grid method is used. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar sweep method, and the nonlinear difference problem is solved by the Newton method. On the basis of an a priori estimate of the solution of a nonlinear difference problem, we prove the convergence of the second degree of Newton’s method.
The numerical calculations carried out show that, for small time intervals, the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%
References
[2] Рысбайулы Б., Адамов А.А. Математическое моделирование тепла и массообменного процесса в многослойном грунте (Монография). // Издательский дом "Қазақ Университетi " – 2020. – С. 211.
[3] Luikov A.V. Heat and Mass Transfer in Capillary-Porous Bodies – 1964.
B. Rysbaiuly, S.D. Alpar 71
[4] Berger J., Dutykh D., Mendes N., Rysbaiuly B. A new model for simulating heat, air and moisture transport in porous
building material // International Journal of Heat and Mass Transfer. – 2019 – Vol. 134. – С. 1041-1060
[5] Tien-Mo Shih, Chao-Ho Sung, Bao Yang. A Numerical Method for Solving Nonlinear Heat Transfer Equations // Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology. – 2008.
[6] Mazumder S. Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods – Academic Press. – 2016.
[7] Lopushansky A., Lopushansky O., Sharyn S. Nonlinear inverse problem of control diffusivity parameter determination for a space-time fractional diffusion equation // Applied Mathematics and Computation. – 2021.
[8] Moore T., Jones M. Solving nonlinear heat transfer problems using variation of parameters // International Journal of Thermal Sciences. – 2015.
[9] Battaglia J-L., Maachou A., Malti R., Melchior P. Nonlinear heat diffusion simulation using Volterra series expansion // International Journal of Thermal Sciences. – 2013.
[10] Nguyen Huy Tuana, Pham Hoang Quanc Some extended results on a nonlinear ill-posed heat equation and remarks on a general case of nonlinear terms // Nonlinear Analysis: Real World Applications. – 2011
[11] Huntul M., Lesnic D. Determination of the time-dependent convection coefficient in two-dimensional free boundary problems. // Engineering Computations. – 2020.
[12] Jumabekova A., Berger J., Dutykh D., Le Meur H. An efficient numerical model for liquid water uptake in porous material and its parameter estimation. // Numerical Heat Transfer, Part A: Applications. – 2019.
[13] Hasanov A. Lipschitz continuity of the Fr?chet gradient in an inverse coefficient problem for a parabolic equation with Dirichlet measured output. // Journal of Inverse and Ill-Posed Problems. – 2018.
[14] Cao K., Lesnic D., Ismailov M. Determination of the time-dependent thermal grooving coefficient. // Journal of Inverse and Ill-Posed Problems. – 2021.
[15] Lesnic D., Hussein M.S., Kamynin V., Kostina B. Direct and inverse source problems for degenerate parabolic equations // Journal of Inverse and Ill-Posed Problems. – 2020.
[16] Kabanikhin S. I., Shishlenin M.A. Theory and numerical methods for solving inverse and ill-posed problems // Journal of Inverse and Ill-Posed Problems. – 2019.
[17] Lesnic D. Inverse Problems with Applications in Science and Engineering // JRC Press, Abingdon, UK. – 2021. – C. 349.
[18] Rysbaiuly B., Rysbaeva N. The method of solving nonlinear heat transfer model in freezing soil. // Eurasian Journal of Mathematical and Computer Applications (EJMCA). – 2020. – Vol.8(4). – С.83-96.
[19] Рысбайулы Б. Обратные задачи нелинейной теплопередачи. // Қазақ Университетi. – 2022. – С.369.
