Transient Heat Conduction Analysis of a Functionally Graded Sphere

Abstract

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.

Keywords

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

Fonksiyonel Derecelendirilmiş Kürenin Geçici Rejimdeki Isı İletim Analizi

Abstract

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.

Keywords

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

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