Recovering of the Heat Transfer Coefficient from the Temperature Measurements

Теория обратных задач используется, чтобы восстановить коэффициент теплопередачи в задачах теплопроводности, используя замеры температуры на границе. Численное решение основано на методе конечных элементов по пространственным переменным, методе конечных разностей по времени и специальной итерационной схемы для определения коэффициента теплопередачи на каждом временном слое. Коэффициент теплопередачи ищется в виде конечного отрезка ряда с неизвестными коэффициентами Фурье, зависящими от времени. Алгоритм решения опирается на теоретические результаты, утверждающие, что задача корректна и сводится к операторному уравнению со сжимающим оператором. Результаты численных экспериментов подтверждают, что задача действительно корректна. Полученные результаты показывают точность, эффективность и надежность предложенного алгоритма. Они устойчивы к случайным возмущениям.

Ключевые слова

обратная задача; коэффициент теплопередачи; параболическое уравнение; тепломассоперенос.

Литература

1. Alifanov O.M. Inverse Heat Transfer Problems. Berlin, Springer, 1994.
2. Tkachenko V.N. Mathematical Modeling, Identification and Control of Technological Processes of Heat Treatment of Materials. Kiev, Naukova Dumka, 2008. (in Russian)
3. Glagolev M.V., Sabrekov A.F. [Determination of Gas Exchange on the Border Between Ecosystem and Atmosphere: Inverse Modelling]. Matematicheskaya biologiya i bioinformatika, 2013, vol. 7, no. 11, pp. 81–101. (in Russian)
4. Borodulin A.I., Desyatkov B.D., Makhov G.A., Sarmanaev S.R. [Determination of Marsh Methane Emission from Measured Values of its Concentration in the Surface Layer of the Atmosphere]. Meteorologiya i gidrologiya, 1997, no. 1, pp. 66–74. (in Russian)
5. Dantas L.V., Orlande H.R.B., Cotta R.M. An Inverse Problem of Parameter Estimation for Heat and Mass Transfer in Capillary Porous Media. International Journal of Heat and Mass Transfer, 2003, vol. 46, no. 9, pp. 1587–1599. DOI: 10.1016/S0017-9310(02)00424-6.
6. Lugon J.Jr., Neto A.J.S. An Inverse Problem of Parameter Estimation in Simultaneous Heat and Mass Transfer in a One-dimensional Porous Medium. Proceedings of COBEM 2003, 17-th International Congress on Mechanical Engineering, San-Paolo, ABCM, 2003, 11 p. Available at: https://abcm.org.br/anais/cobem/2003/html/pdf/COB03-1158.pdf
7. Cao K., Lesnic D., Colaco M.J. Determination of Thermal Conductivity of Inhomogeneous Orthotropic Materials from Temperature Measurements. Inverse Problems in Science and Engineering, 2018, vol. 27, no. 10, pp. 1372–1398. DOI: 10.1080/17415977.2018.1554654
8. Varan L.A.B., Orlande H.R.B., Vianna F.L.V. Estimation of the Convective Heat Transfer Coefficient in Pipelines with the Markov Chain Monte–Carlo Method. Blucher Mechanical Engineering Proceedings, 2014, vol. 1, no. 1, pp. 1214–1225. DOI: 10.5151/meceng-wccm2012-18647sthash.z1ehN7bo.dpuf
9. Osman A.M., Beck J.V. Nonlinear Inverse Problem for the Estimation of Time-and-Space-Dependent Heat-Transfer Coefficients. Journal of Thermophysics and Heat Transfer, 2003, vol. 3, no. 2, pp. 146–152. DOI: 10.2514/3.141
10. Colac M.J., Orlande H.R.B. Inverse Natural Convection Problem of Simultaneous Estimation of Two Boundary Heat Fluxes in Irregular Cavities. Journal of Thermophysics and Heat Transfer, 2004, vol. 47, no. 6, pp. 1201–1215. DOI: 10.1016/j.ijheatmasstransfer.2003.09.007
11. Avallone F., Greco C.S., Ekelschot D. 2D-Inverse Heat Transfer Measurements by IR Thermography in Hypersonic Flows. Proceedings of the 11-th International Conference on Quantitative InfraRed Thermography, Naples, 2012, pp. 1–13.
12. Farahani S.D., Kowsary F., Ashjaee M. Experimental Estimation Heat Flux and Heat Transfer Coefficient by Using Inverse Methods. Scientia Iranica B, 2016, vol. 3, no. 4, pp. 1777–1786. DOI: 10.24200/sci.2016.3925
13. Su Jian, Hewitt G.F. Inverse Heat Conduction Problem of Estimating Time-Varying Heat Transfer Coefficient. Numerical Heat Transfer Applications, 2004, vol. 45, no. 8, pp. 777–789. DOI: 10.1080/1040778049042499
14. Hao Dinh Nho, Thanh Phan Xua, Lesnic D. Determination of the Heat Transfer Coefficients in Transient Heat Conduction. Inverse Problems, 2013, vol. 29, article ID: 095020, 21 p. DOI: 10.1088/0266-5611/29/9/095020
15. Onyango T.M., Ingham D.B., Lesnic D. Restoring Boundary Conditions in Heat Conduction. Journal of Engineering Mathematics, 2008, vol. 62, no. 1, pp. 85–101. DOI: 10.1007/s10665-007-9192-0
16. Wang Shoubin, Zhang Li, Sun Xiaogang, Jial Huangchao. Solution to Two-Dimensional Steady Inverse Heat Transfer Problems with Interior Heat Source Based on the Conjugate Gradient Method. Mathematical Problems in Engineering, 2017, vol. 2017, article ID: 2861342, 9 p. DOI: 10.1155/2017/2861342
17. Sladek J., Sladek V., Wen P.H., Hon Y.C. The Inverse Problem of Determining Heat Transfer Coefficients by the Meshless Local Petrov–Galerkin Method. Computer Modeling in Engineering and Sciences, 2009, vol. 48, no. 2, pp. 191–218. DOI: 10.3970/cmes.2009.048.191
18. Jin Bangti, Lu Xiliang. Numerical Identification of a Robin Coefficient in Parabolic Problems. Mathematics of Computation, 2012, vol. 81, no. 279, pp. 1369–1398.
19. Rundell W., Yin Hong-Ming. A Parabolic Inverse Problem with an Unknown Boundary Condition. Journal of Differential Equations, 1990, vol. 86, no. 2, pp. 234–242. DOI: 10.1016/0022-039690)90031-J
20. Pilant M., Rundell W. An Iteration Method for the Determination of an Unknown Boundary Condition in a Parabolic Initial-Boundary Value Problem. Proceedings of the Edinburgh Mathematical Society, 1989, vol. 32, no. 1, pp. 59–71. DOI: 10.1017/S001309150000691X
21. Da Silva W.B.J., Dutra C.S., Kopperschmidt C.E.P., Lesnic D., Aykroyd R.G. Sequential Particle Filter Estimation of a Time-Dependent Heat Transfer Coefficient in a Multidimensional Nonlinear Inverse Heat Conduction Problem. Applied Mathematical Modelling, 2012, vol. 89, no. 1, pp. 654–668.
22. Da Silva W.B.J., Dutra C.S., Kopperschimidt C.E.P., Lesnic D., Aykroyd R.G. Sequential Estimation of the Time-Dependent Heat Transfer Coefficient Using the Method of Fundamental Solutions and Particle Filters. Inverse Problems in Science and Engineering, 2021, vol. 29, no. 13, pp. 3322–3341.
23. Hao Dinh Nho, Huong Bui Viet, Thanh Phan Xua, Lesnic D. Identification of Nonlinear Heat Transfer Laws from Boundary Observations. Applicable Analysis, 2015, vol. 94, no. 9, pp. 1784–1799. DOI: 10.1080/00036811.2014.948425
24. Slodicka M., Van Keer R. Determination of a Robin Coefficient in Semilinear Parabolic Problems by Means of Boundary Measurements. Inverse Problems, 2022, vol. 18, no. 1, pp. 139–152. DOI: 10.1088/0266-5611/18/1/310
25. Rosch A. Stability Estimates for the Identification of Nonlinear Heat Transfer Laws. Inverse Problems, 1996, vol. 12, no. 5, pp. 743–756. DOI: 10.1088/0266-5611/12/5/015
26. Knupp D.C., Abreu L.A.S. Explicit Boundary Heat Flux Reconstruction Employing Temperature Measurements Regularized via Truncated Eigenfunction Expansions. International Communications in Heat and Mass Transfer, 2016, vol. 78, pp. 241–252. DOI: 10.1016/j.icheatmasstransfer.2016.09.012
27. Kolesnik S.A., Formalev V.F., Kuznetsova E.L. The Inverse Boundary Thermal Conductivity Problem of Recovery of Heat Fluxes to the Boundries of Anisotropic Bodies. High Temperature, 2015, vol. 53, no. 1, pp. 68–72. DOI: 10.7868/S0040364413050062
28. Alghamdi S.A. Inverse Estimation of Boundary Heat Flux for Heat Conduction Model. Journal of King Abdulaziz University: Engineering Sciences, 2010, vol. 21, no. 1, pp. 73–95. DOI: 10.4197/ENG.21-1.5
29. Kostin A.B., Prilepko A.I. Some Problems of Restoring the Boundary Condition for a Parabolic Equation. II. Differential Equations, 1996, vol. 32, no. 11, pp. 1515–1525.
30. Pyatkov S.G., Baranchuk V.A. Determination of the Heat Transfer Coefficient in Mathematical Models of Heat and Mass Transfer. Mathematical Notes, 2023, vol. 113, no. 1, pp. 93–108. DOI: 10.1134/S0001434623010108
31. Duda P. Solution of Multidimensional Inverse Heat Conduction Problem. Heat and Mass Transfer, 2003, vol. 40, no. 1, pp. 115–122. DOI: 10.1007/s00231-003-0426-z
32. Maciejewska B. The Application of Beck’s Method Combined with Fem and Trefftz functions to Determine the Heat Transfer Coefficient in a Minichannel. Journal of Theoretical and Applied Mechanics, 2017, vol. 55, no. 1, pp. 103–116. DOI: 10.15632/jtam-pl.55.1.103
33. Denk R., Hieber M., Pruss J. Optimal Lp - Lq-estimates for parabolic boundary value problems with inhomogeneous data. Mathematische Zeitschrift, 2007, vol. 257, no. 1, pp. 193–224. DOI: 10.1007/s00209-007-0120-9
34. Triebel H. Interpolation Theory, Functional Spaces, Differential Operators. North-Holland, Amsterdam, 1978.
35. Amann H. Compact Embeddings of Vector-Valued Sobolev and Besov spaces. Glasnik Matematicki, 2000, vol. 35, no. 1, pp. 161–177.
36. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Linear and Quasilinear Equations of Parabolic Type. Providence, American Mathematical Society, 1968.