Abstract
This paper presents the solution of the theoretical model of heat conduction based on timefractional Fourier equation for a finite hollow cylinder treated with heat flux on one of the front surfaces. A derivative of fractional order in the Caputo sense was applied to record the temperature derivative in time. The distributions of temperature fields in the hollow cylinder were determined with the use of Fourier-Bessel series, as surface functions of two variables (r,θ) . The distributions of temperature fields were determined using analytical methods and applying integral transformation methods. The Laplace transform with reference to time, the Fourier finite cosine transform with reference to axial coordinatezand Marchi-Zgrablich transform for radial coordinater. The fractional heat conduction equation was analysed for 0 < α ≤ 2
Highlights
The research papers concerning heat conduction in structural elements in machines and devices are based on the classical Fourier’s law
The classical theory of heat conduction is based on the Fourier law, which might be recorded in the following form: q g r a d ( )
T q t C p t where is density and Cp is the specific heat capacity at constant pressure. Another aspect of the research papers which has gained in popularity over the last thirty years is differential and integral calculus conducted on derivatives of fractional order [8,9,10,11,12,13,14,15,16]
Summary
The research papers concerning heat conduction in structural elements in machines and devices are based on the classical Fourier’s law. They might be generally classified e.g. in terms of the shape of the medium where the heat conduction is considered. Where is density and Cp is the specific heat capacity at constant pressure Another aspect of the research papers which has gained in popularity over the last thirty years is differential and integral calculus conducted on derivatives of fractional order [8,9,10,11,12,13,14,15,16]. With reference to the subject of this paper, the results of heat conduction in the hollow cylinder under Gaussiandistributed heat flux were presented
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