Publikacje

Streszczenie:

Cel
W celu zmniejszenia obciążenia cieplnego łopatki turbiny gazowej jej powierzchnia pokryta jest zewnętrzną warstwą ceramiki o wysokiej odporności termicznej. Istotą problemu jest dobór ceramiki o tak niskim współczynniku przewodzenia ciepła i grubości, aby nie przekraczać dopuszczalnej temperatury metalu na styku metal-ceramika ze względu na utratę właściwości mechanicznych.
Projekt / metodologia / podejście
Dlatego dla zadanych zmian temperatury w czasie na styku metal-ceramika, zmian temperatury w czasie na wewnętrznej stronie ostrza oraz założonej temperatury początkowej, należy wyznaczyć zmianę temperatury w czasie na zewnętrznej powierzchni ceramiki. Przedstawiony w ten sposób problem jest problemem typu Cauchy'ego. Analizując problem, bierze się pod uwagę, że właściwości termofizyczne metalu i ceramiki mogą zależeć od temperatury. Ze względu na cienką warstwę ceramiki w stosunku do grubości ścianki problem rozpatrywany jest w obszarze w warstwie płaskiej. W związku z tym rozważany jest jednowymiarowy niestacjonarny przepływ ciepła.
Wyniki
Zbadano zakres stabilności problemu Cauchy'ego w funkcji kroku czasowego, grubości ceramiki oraz właściwości termofizyczne metalu i ceramiki. Obliczenia numeryczne obejmowały również wpływ zaburzeń temperatury na granicy faz metal-ceramika na rozwiązanie problemu odwrotnego.
Praktyczne implikacje
Model obliczeniowy można wykorzystać do analizy przepływu ciepła w łopatkach turbiny gazowej z barierą termiczną.
Oryginalność / wartość
W literaturze przedstawiono szereg zagadnień odwrotnych tego typu, które są rozważane w artykule. Zagadnienia odwrotne, zwłaszcza te typu Cauchy'ego, są źle uwarunkowane numerycznie, co oznacza, że niewielka zmiana danych wejściowych może skutkować znacznymi błędami rozwiązania. W takim przypadku potrzebna jest regularyzacja odwrotnego problemu. Przedstawiony w artykule problem Cauchy'ego nie wymaga jednak regularyzacji.

Abstract:

Purpose
In order to reduce the heat load of a gas turbine blade, its surface is covered with an outer layer of ceramics with high thermal resistance. The essence of the problem is the selection of ceramics with such a low heat conduction coefficient and thickness, so that the permissible metal temperature is not exceeded on the metal-ceramics interface due to the loss of mechanical properties.
Design/methodology/approach
Therefore, for given temperature changes over time on the metal-ceramics interface, temperature changes over time on the inner side of the blade and the assumed initial temperature, the temperature change over time on the outer surface of the ceramics should be determined. The problem presented in this way is a Cauchy type problem. When analyzing the problem, it is taken into account that thermophysical properties of metal and ceramics may depend on temperature. Due to the thin layer of ceramics in relation to the wall thickness, the problem is considered in the area in the flat layer. Thus, a one-dimensional nonstationary heat flow is considered.
Findings
The range of stability of the Cauchy problem as a function of time step, thickness of ceramics and thermophysical properties of metal and ceramics are examined. The numerical computations also involved the influence of disturbances in the temperature on metal-ceramics interface on the solution to the inverse problem.
Practical implications
The computational model can be used to analyze the heat flow in gas turbine blades with thermal barrier.
Originality/value
A number of inverse problems of the type considered in the paper are presented in the literature. Inverse problems, especially those Cauchy-type, are ill-conditioned numerically which means that a small change in the inputs may result in significant errors of the solution. In such a case, regularization of the inverse problem is needed. However, the Cauchy problem presented in the paper does not
require regularization.

B I B L I O G R A F I AAlifanov O.M., Inverse heat-transfer problems in the investigation of thermal processes and the design of engineering systems. J Eng Phys 1977. https://doi.org/10.1007/BF00865370
Alifanov, O. M. (1994), Inverse heat transfer problems, Springer-Verlag, ISBN 0-387-53679-5, New York
Bakushinskii A., Goncharsky A., Ill-Posed Problems: Theory and Applications. Kluwer, Dordrecht, 1995
Caillé L, Marin L, Delvare F. A meshless fading regularization algorithm for solving the Cauchy problem for the three-dimensional Helmholtz equation. Numer Algorithms 2019
82:869–94. https://doi.org/10.1007/s11075-018-0631-y.
Ciałkowski MJ, Frąckowiak A, Heat Functions and Their Application for Solving Heat Transfer and Mechanical Problems, Poznań University of Technology Publishers, Poznań, Poland (in Polish), 2000
De Lillo S., Lupo G .and Sanchini G, A Cauchy problem in nonlinear heat conduction. J. Phys. A: Math. Gen. 39, 7299-7304, 2006
Engl H.W., Hanke M., Neubauer A., Regularization of Inverse Problems. Kluwer Academic Publisher, Dordrecht Boston London, 2000.
Frąckowiak, A. and Ciałkowski, M., Application of discrete Fourier transform to inverse heat conduction problem regularization, Int. J. of Num. Meth. for Heat and Fluid Flow, 28, 1, 239-253, 2018.
Frąckowiak, A. and Ciałkowski, M., Application of discrete Fourier transform to inverse heat conduction problem regularization, Int. J. of Num. Meth. for Heat and Fluid Flow, 28, 1, 239-253, 2018.
Frąckowiak, A., Ciałkowski, M. and Wróblewska, A., Application of iterative algorithms for gas-turbine blades cooling optimization, Int. J. of Thermal Sciences, 118, 198-206, 2017.
Frąckowiak, A., Spura, D., Gampe, U. and Ciałkowski, M., Determination of heat transfer coefficient in a t-shaped cavity by means of solving the inverse heat conduction problem, Int. J. of Num. Meth for Heat and Fluid Flow, 2019, DOI: 10.1108/HFF-09-2018-0484.
Frąckowiak, A., von Wolfersdorf, J. and Ciałkowski, M., Optimization of cooling of gas turbine blades with channels filled with porous material, Int. J. of Thermal Sciences, 136, 370-378. 2019
Gockenbach M.S., Linear Inverse Problems and Tikhonov Regularization Series: Carus Mathematical Monographs (Book 32), 2016
Grysa K., Hożejowska S., Maciejewska B., Adjustment calculus and Trefftz functions applied to local heat transfer coefficient determination in a minichannel, J. of Theor. and Appl. Mech., 50, 4, 1087-1096, 2012
Grysa K., Maciąg A., Adamczyk-Krasa J., Trefftz Functions applied to Direct and Inverse Non-Fourier Heat Conduction Problems, J. of Heat Transfer -Trans of the ASME, 136, 9,1-9, 2014.
Grysa K., Maciąg A., Walaszczyk M., Cebo-Rudnicka A., Identification of the heat transfer coefficient during cooling process by means of Trefftz method, Engng Anal. with Boundary Elements, 95, 10, 33-39, 2018.
Grysa K., Trefftz functions and their applications in solving the inverse problems. Kielce University of Technology Publishers, 2010 (in Polish).
Hadamard J., Sur les problèmes aux dérivées partielles et leur signification physique. Princeton. 1902.
Haji-Sheikh A., Beck J.V., Agonafer D., Steady-state heat conduction in multi-layer bodies. Int J Heat Mass Transf 2003
46:2363–79. https://doi.org/10.1016/S0017-9310(02)00542-2.
Haò D.N., A Noncharacteristic Cauchy Problem for Linear Parabolic Equations I: Solvability, Mathematische Nachrichten, 171, 177–206, 1995.
Joachimiak M., and Ciałkowski M., Stable solution to non-stationary inverse heat conduction equation. Arch. of Thermodynamics 2018
39:25–37.
Joachimiak M., Frąckowiak A., and Ciałkowski M., Solution of inverse heat conduction equation with the use of Chebyshev polynomials. Arch. of Thermodynamics 2016
37:73–88.
Joachimiak M., Joachimiak D., Ciałkowski M., Małdziński L., Okoniewicz P., Ostrowska K., Analysis of the heat transfer for processes of the cylinder heating in the heat-treating furnace on the basis of solving the inverse problem, Int. J. of Thermal Sciences, 145, 1-11,2019.
Kurpisz K., Nowak A.J., Inverse Thermal Problems, Computational Mechanics Publications, Southampton, UK and Boston, USA, 1995.
Liu J.C., Wei T., The method of lines to reconstruct a moving boundary for a one-dimensional heat equation in a multilayer domain. J. Eng. Math.
71:157–70, 2011. https://doi.org/10.1007/ s10665-010-9430-8.
Liu J.C., Wei T.. The method of lines to reconstruct a moving boundary for a one-dimensional heat equation in a multilayer domain. J Eng Math 2011. https://doi.org/10.1007/s10665-010-9430-8.
Macia̧g A, Al-Khatib M.J., Stability of solutions of the overdetermined inverse heat conduction problems when discretized with respect to time. Int. J Numer Methods Heat Fluid Flow 2000.
Maciag A, Trefftz Functions for Some Direct and Inverse Problems of Mechanics, Kielce, University of Technology Publishers, Kielce, Poland (in Polish), 2009.
Maciąg A., Grysa K., Temperature dependent thermal conductivity determination and source identification for nonlinear heat conduction by means of the Trefftz and homotopy perturbation methods, Int. J. of Heat and Mass Transfer, 100,627-633, 2016.
Marin L. An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation. Comput Mech 2010. https://doi.org/10.1007/s00466-010-0480-6.
Marin L., Lesnic D., The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Computers & Structures 83(4-5):267-278, 2005, DOI: 10.1016/j.compstruc.2004.10.005
Ramm A.G., Inverse problems. Springer, 2004.
Simões N., Simões I., Tadeu A., Vasconcellos C.A.B., Mansur W.J., 3D transient heat conduction in multilayer systems - Experimental validation of semi-analytical solution. Int. J. Therm. Sci. 2012
57:192–203. https://doi.org/10.1016/j.ijthermalsci. 2012.02.007.
Tikhonov A.N. and Arsenin V.Y. Solution of Ill-Posed Problems. New York: Wiley,1977.
Xiong X.T., Hon Y.C., Regularization error analysis on a one-dimensional inverse heat conduction problem in multilayer domain. Inverse Probl. Sci. Eng. 2013
21:865–87. https://doi.org/10.1080/17415977.2013.788168.
Yang F., Zhang P., Li X.X., The truncation method for the Cauchy problem of the inhomogeneous Helmholtz equation. Appl Anal 2019
98:991–1004. https://doi.org/10.1080/00036811. 2017.1408080.

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Cauchy type nonlinear inverse problem in a two-layer area