Transient Heat Conduction Analysis of a Functionally Graded Sphere

Öz

The functionally graded thick-walled hollow sphere is analysed in transient regime using a hyperbolic heat conduction model. It is assumed that the material properties, such as heat conduction coefficient, density, and specific heat change exponentially in radial direction. Under these conditions, a partial differential equation which is difficult to solve by conventional methods is obtained. By applying Laplace transform to this differential equation, time-independent linear ordinary differential equation is created in Laplace space. Then the differential equation is solved numerically using Chebyshev Pseudospectral Method and the solution in time space is obtained by using the Modified Durbin Inverse Transformation Method. The dynamic reactions of heat and heat flux in the transient regime and their behaviour against thermal relaxation, relative temperature parameters are investigated for a special material of metal-ceramic mixture and comparisons are made with homogeneous material. The solutions available in the literature are used to validate the results. The combined method used in this study is shown to be a well-structured, simple, and effective.

Anahtar Kelimeler

• Hyperbolic Heat Conduction
• Functionally Graded Sphere
• Laplace Transform
• Chebyshev Pseudospectral Method
• Durbin’s Inverse Method

Fonksiyonel Derecelendirilmiş Kürenin Geçici Rejimdeki Isı İletim Analizi

Öz

Fonksiyonel derecelendirilmiş kalın cidarlı içi boş kürenin hiperbolik ısı iletim modeli kullanılarak geçici rejimdeki analizi ele alınmıştır. Isı iletim katsayısı, yoğunluk ve özgül ısı gibi malzeme özelliklerinin radyal yönde üstel olarak değiştiği kabul edilmiştir. Bu koşullar altında, geleneksel analitik yöntemlerle çözülmesi zor olan değişken katsayılı kısmi diferansiyel denklem elde edilir. Bu diferansiyel denkleme Laplace dönüşümü uygulanarak, Laplace uzayında zamandan bağımsız lineer adi diferansiyel denklem oluşturulur. Daha sonra, 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 termal gevşeme, bağıl sıcaklık parametrelerine karşı geçici rejimdeki dinamik davranışları metal-seramik karışımı özel bir malzeme için incelenmiş, homojen malzeme ile karşılaştırmalar yapılmıştır. Literatürde mevcut olan çözümler, bu çalışmada elde edilen sonuçları doğrulamak için kullanılmıştır. Bu çalışmada kullanılan birleştirilmiş yöntemin, iyi yapılandırılmış, basit, etkili bir yöntem olduğu gösterilmiştir.

Anahtar Kelimeler

• Hiperbolik Isı İletimi
• Fonksiyonel Derecelendirilmiş Küre
• Laplace Dönüşümü
• Chebyshev Pseudospektral Yöntemi
• Durbin Ters Dönüşüm Yöntemi

Kaynakça

1. Yamanouchi, M., Koizumi, M., Hirai, T. & Shiota, I. (1990). Proceedings of the 1st International Symposium on Functionally Gradient Materials, Sendai, Japan.
2. Koizumi, M. (1993). The Concept of FGM. Ceramic Transactions, Functionally Gradient Materials, 34, 3-10.
3. Wilhelm, H. E. & Choi, S. H. (1975). Nonlinear hyperbolic theory of thermal waves in metals. The Journal of Chemical Physics, 63(5), 2219-2123.
4. Chen, H. T. & Lin, J. Y. (1993). Numerical analysis for hyperbolic heat conduction. International Journal of Heat and Mass Transfer, 36(11), 2891-2898.
5. Lin, J. Y. & Chen, H. T. (1994). Numerical solution of hyperbolic heat conduction in cylindrical and spherical systems. Applied Mathematical Modelling, 18, 384-390.
6. Antaki, P. J. (1995). Key Features of analytical solutions for hyperbolic heat conduction, American Institute of Aeronautics and Astronautics, 95(2044), 1-15.
7. Zanchini, E. & Pulvirenti, B. (1998). Periodic heat conduction with relaxation time in cylindrical geometry, Heat and Mass Transfer, 33, 319-326.
8. Al-Nimr, M. A. & Naji, M. (2000). The hyperbolic heat conduction equation in ananisotropic material. International Journal of Thermophysics, 21(1), 281-287.

9. Chen, H. T., Peng, S. Y. & Yang, P. C. (2001). Numerical method for hyperbolic inverse heat conduction problems. International Communications in Heat and Mass Transfer, 28(6), 847-856.
10. Jiang, F. (2005). Analytical solution for hyperbolic heat conduction in a hollow sphere, Journal of Thermophysics and Heat Transfer, 19(4), 595-598.
11. Tsai, C. S., Lin, Y. C. & Hung, C. I. (2005). A study on the non-Fourier effects in spherical media due to sudden temperature changes on the surfaces. Heat Mass Transfer, 41, 709-716.
12. Jiang, F. (2006).Solution and analysis of hyperbolic heat propagation in hollow spherical objects. Heat Mass Transfer, 42, 1083-1091.
13. Hosseini, S. M., Akhlaghi, M. & Shakeri, M. (2007). Transient heat conduction in functionally graded thick hollow cylinders by analytical method. Heat Mass Transfer, 43, 669-675.
14. Babaei, M. H. & Chen, Z. T.(2008). Hyperbolic heat conduction in a functionally graded hollow sphere. International Journal of Thermophysics, 29, 1457-1469.
15. Moosaie, A. (2009). Axisymmetric non-Fourier temperature field in a hollow sphere. Archive of Applied Mechanics, 79, 679-694.
16. Shirmohammadi, A. & Moosaie, A. (2009). Non-Fourier heat conduction in a hollow sphere with periodic surface heat flux. International Communications in Heat and Mass Transfer, 36, 827-833.
17. Keles, I. & Conker, C. (2011). Transient hyperbolic heat conduction in thick-walled FGM cylinders and spheres with exponentially-varying properties, European Journal of Mechanics A/Solids, 30, 449-455.
18. 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.
19. Trefethen, L. N. (2000). Spectral Methods in Matlab. SIAM, Philadelphia, PA.
20. Eker, M., Yarımpabuç, D. & Çelebi, K. (2020). Thermal stress analysis of functionally graded solid and hollow thick-walled structures with heat generation. Engineering Computations, 38(1), 371-391.
21. Eker, M., Yarımpabuç, D., Yıldırım, A. & Çelebi, K. (2020). Elastic solutions based on the Mori-Tanaka scheme for pressurized functionally graded cylinder. Journal of Applied Mathematics and Computational Mechanics, 19(4), 57-68.
22. Durbin, F. (1974). Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate's method. The Computer Journal, 17(4), 371-376.
23. Dubner, R. & Abate, J. (1968). Numerical inversion of Laplace transforms by relating them to the finite Fourier Cosine transform. Journal of Applied and Computational Mechanics, 15(1), 115-123.
24. Çalım, F. F. (2009). Dynamic analysis of beams on viscoelastic foundation. European Journal of Mechanics- A/Solids, 28(3), 469-476.
25. Temel, B., Yıldırım, S. & Tutuncu, N. (2014). Elastic and viscoelastic response of heterogeneous annular structures under arbitrary transient pressure. International Journal of Mechanical Sciences, 89, 78-83.
26. Çalım, F. F. (2016). Transient analysis of axially functionally graded Timoshenko beams with variable cross-section. Composites Part B: Engineering, 98, 472-483.
27. Temel, B. & Şahan, M. F. (2018). Investigation of the Efficiency of the Solution of a Simple Mechanical Model by Using Laplace Transformation. American Journal of Engineering Research (AJER), 7(10), 276-282.
28. Noori, A. R., Aslan, T. A. & Temel, B. (2021). Dynamic Analysis of Functionally Graded Porous Beams Using Complementary Functions Method in the Laplace Domain. Composite Structures, 256, 113094.
29. Temel, B., Aslan, T. A. & Noori, A. R. (2021). In-plane vibration analysis of parabolic arches having a variable thickness. Int. J. Dynam. Control.
30. Narayan, G. V. (1979). Numerical operational methods in structural dynamics. Doktora Tezi, University of Minnesota, Mineapolis, MN.
31. Yildirim, A., Yarimpabuç, D. & Celebi K. (2020). Transient thermal stress analysis of functionally graded annular fin with free base. Journal of Thermal Stresses, 43(9), 1138-1149.