Fonksiyonel Derecelendirilmiş Silindirin Isı İletiminin Chebyshev Pseudospektral Yöntemi ile Geçici Rejimdeki Analizi

Fourier olmayan hiperbolik ısı iletim modeli kullanılarak fonksiyonel derecelendirilmiş sonsuz uzunlukta içi boş silindirin geçici rejimdeki analizi yapılmıştır. Termal gevşeme süresi dışında malzeme özelliklerinin radyal yönde üstel olarak değiştiği kabul edilmiştir. Sonsuz uzunlukta bir silindir ele alındığından silindirin alt ve üst ucundaki etkiler ihmal edilmiştir. Bu koşullar altında, uzay yönünde sadece radyal değişkene bağlı değişken katsayılı kısmi diferansiyel denklem elde edilir. Bu diferansiyel denkleme Laplace dönüşümü uygulanarak, Laplace uzayında zamandan bağımsız elde edilen lineer adi diferansiyel denklem, Chebyshev Pseudospektral yöntemi kullanılarak sayısal olarak çözülüp, Modifiye Edilmiş Durbin Ters Dönüşüm Yöntemi kullanılarak zaman uzayındaki çözüm elde edilir. Sıcaklık ve ısı akısının geçici dinamik tepkileri, özel bir metal-seramik karışımı alınarak ve çeşitli göreceli sıcaklık değişikliklerine göre incelenmiştir. Farklı zamanlarda sıcaklık dağılımı ve ısı akısının davranışı farklı şekiller üzerinde gösterilmiştir. Bu birleştirilmiş yöntemin doğruluğunu göstermek için literatürde mevcut olan çözümler kullanılmıştır. Bu çalışmada kullanılan birleştirilmiş yöntemin, iyi yapılandırılmış, basit, etkili bir yöntem olduğu gösterilmiştir.

Heat Conduction Analysis of a Functionally Graded Cylinder in Transient Regime with Chebyshev Pseudospectral Method

Transient analysis of functionally graded infinite length hollow cylinder is performed using non-Fourier hyperbolic heat conduction model. Except the thermal relaxation time, the material properties are assumed to change exponentially in the radial direction. Since an infinite-length cylinder is considered, the effects at the upper and lower ends of the cylinder are neglected. Under these conditions, a partial differential equation with variable coefficients is obtained which is only depend on the radial variable in the space direction. By applying Laplace transform to this differential equation, the linear ordinary differential equation obtained in Laplace space is solved numerically by using Chebyshev Pseudospectral method and the solution in time space is obtained by using Modified Durbin Inverse Transformation Method. The transient dynamic responses of temperature and heat flux are investigated by taking a special metal-ceramic mixture and examining various relative temperature changes. The behavior of temperature distribution and heat flux at different times is shown in the form of graphs. The solutions available in the literature are used to demonstrate the accuracy of this combined method. The combined method used in this study has been shown to be a well structured, simple, effective.

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[1] Birman V., Byrd L.W. 2007. Modeling and analysis of functionally graded materials and structures. Applied Mechanics Reviews, 60 (5): 195–216.
[2] Shen M., Bever M. 1972. Gradients in polymeric materials. Journal of Materials science, 7 (7): 741-746.
[3] Niino M., Hirai T., Watanabe R. 1987. The functionally gradient materials. Journal of the Japan Society for Composite Materials, 13 (1): 257-264.
[4] Babaei M.H., Chen Z.T. 2010. Transient hyperbolic heat conduction in a functionally graded hollow cylinder. Journal Of Thermophysics and Heat Transfer, 24 (2): 325-330.
[5] Wilhelm H.E., Choi S.H. 1975. Nonlinear hyperbolic theory of thermal waves in metals. The journal of Chemical Physics, 63 (5): 2119-2123.
[6] Chen H.T., Lin J.Y. 1993. Numerical analysis for hyperbolic heat conduction. International Journal of Heat and Mass Transfer, 36 (11): 2891-2898.
[7] Lin J.Y., Chen H.T. 1994. Numerical solution of hyperbolic heat conduction in cylindrical and spherical systems. Applied Mathematical Modelling, 18 (7): 384-390.
[8] Antaki P.J. 1995. Key features of analytical solutions for hyperbolic heat conduction. AIAA Paper, 95: 2044.
[9] Zanchini E., Pulvirenti B. 1998. Periodic heat conduction with relaxation time in cylindrical geometry. Heat and Mass Transfer, 33 (4): 319-326.
[10] Al–Nimr M. A., Naji M. 2000. The hyperbolic heat conduction equation in an anisotropic material. International Journal of Thermophysics, 21 (1): 281-287.
[11] Chen H.T., Peng S.Y., Yang P.C. 2001. Numerical method for hyperbolic inverse heat conduction problems. Int. Comm. Heat Mass Transfer, 28 (6): 847-856.
[12] Tarn J.Q., Wang Y.M. 2003. Heat conduction in a cylindrically anisotropic tube of a functionally graded material. Chinese Journal of Mechanics (Series A), 19: 365-372.
[13] Liu K.C., Lin C.N., Wang J.S. 2005. Numerical solutions for the hyperbolic heat conduction problems in a layered solid cylinder with radiation surface. Applied Mathematics and Computation, 164: 805-820.
[14] Lu X., Tervola P., Viljanen M. 2006. Transient analytical solution to heat conduction in composite circular cylinder. International Journal of Heat and Mass Transfer, 49: 341-348.
[15] Hosseini S.M., Akhlaghi M., Shakeri M. 2007. Transient heat conduction in functionally graded thick hollow cylinders by analytical method. Heat and Mass Transfer, 43 (7): 669-675.
[16] Darabseh T., Naji M., Al-Nimr M. 2008. Transient thermal stresses in an orthotropic cylinder under the hyperbolic heat conduction model. Heat Transfer Engineering, 29 (7): 632-642.
[17] Chen T.M. 2010. Numerical solution of hyperbolic heat conduction problems in the cylindrical coordinate system by the hybrid Green’s function method. International Journal of Heat and Mass Transfer, 53: 1319-1325.
[18] Keles I., Conker C. 2011. Transient hyperbolic heat conduction in thick-walled FGM cylinders and sphereswith exponentially-varying properties. European Journal of Mechanics A/Solids, 30 (1): 449-455.
[19] Afshin A., Nejad M.Z, Dastani K. 2017. Transient thermoelastic analysis of FGM rotating thick cylindrical pressure vessels under arbitrary boundary and initial conditions. Jornal of Computational Applied Mechanics, 48 (1): 15-26.
[20] Gottlieb D., Hussaini M.Y., Orszag S.A. 1984. Introduction: Theory and Applications of Spectral Methods. SIAM, USA.
[21] Voigt R.G., Gottlieb D., Hussaini M.Y. 1984. Spectral methods for partial differential equations. SIAM, USA.
[22] Trefethen L.N. 2000. Spectral methods in matlab. SIAM, USA.
[23] Durbin F. 1974. Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method. The Computer Journal, 17 (4): 371-376.
[24] Nowinski J. N. 1978. Theory of thermoelasticity with applications. Sijthoff-Noordhoff, Alphen aan den Rijn, Netherlands.
[25] Narayan G.V. 1979. Numerical operational methods in structural dynamics. Ph.D.thesis, University of Minnesota, Mineapolis, MN.
[26] Dubner R., Abate J. 1968. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. Journal Applied and Computational Mechanics, 15 (1): 115- 123.
[27] Yıldırım A., Yarımpabuç D., Celebi K. 2020. Transient thermal stress analysis of functionally graded annular fin with free base. Journal of Thermal Stresses, 43 (9): 1138–1149.