Temperature solution in multi-dimensional multi-layer bodies

Haji‐Sheikh, A.; Beck, James V.

doi:10.1016/s0017-9310(01)00279-4

Cited by 150 publications

(67 citation statements)

References 10 publications

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“…If fractional order  is in the interval (1,2], then the initial condition for the derivative i T t   is also required [22].…”

Section: T R T a T T T R T Rmentioning

confidence: 99%

“…Heat conduction problems in layered slabs, layered cylinders, and layered spheres modelled according to Fourier's law by a parabolic differential equation have been considered by many authors, for example in [1][2][3], where analytical solutions to the problems in the form of eigenfunction expansions were presented. Heat conduction in layered bodies in spherical coordinates was recently investigated in [4][5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

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An analytical solution to the problem of time-fractional heat conduction in a composite sphere

Kukla

Siedlecka

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative. analytical solution to the problem for thermal and mechanical properties of the sphere was obtained in the form of power functions of the radial direction. Thermal stresses in a sphere of a functionally graded material were also considered by Pawar et al. [9]. Transient temperature distribution was determined by assuming that the material properties of the sphere were exponential functions of the radial direction. The parabolic differential equation of heat conduction, derived under the framework of the classical theory of heat conduction is based on the local Fourier law. Non-local generalizations of the Fourier law lead to non-classical theories, in which the parabolic equation is replaced by a time-fractional and/or space-fractional heat conduction differential equation [10]. In these fractional differential equations, different kinds of derivatives of fractional order are used (the Riemann-Liouville derivative, the Caputo derivative and the Grűnwald-Letnikov derivative). Moreover, the boundary conditions may also include the fractional derivatives. The fundamentals of fractional calculus and of the theory of fractional differential equations are given in [11][12][13][14][15]. Some applications of fractional order calculus to modelling of real-world phenomena are presented in [16][17][18].Heat conduction problems formulated under the framework of the non-classical theories in the spherical coordinates with fractional Caputo or Riemann-Liouville derivatives were studied in [19,20]. An approximate analytical solution of time-fractional heat conduction in a composite medium consisting of an infinite matrix and a spherical inclusion is presented by Povstenko in [19]. The perfect thermal contact was realized by the conditions of equality of temperatures and heat fluxes at the boundary surfaces, wherein the heat fluxes are expressed by a Riemann-Liouville fractional derivative. An analytical solution to the problem of time-fractional heat conduction in a multilayered slab was presented by Siedlecka and Kukla in [20]. Ning and Jiang [21] use the Laplace transform and the method of variable separation to determine an analytical

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“…If fractional order  is in the interval (1,2], then the initial condition for the derivative i T t   is also required [22].…”

Section: T R T a T T T R T Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An analytical solution to the problem of time-fractional heat conduction in a composite sphere

Kukla

Siedlecka

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…Applying the Laplace transformation to equation (1) and using the property (10), we obtain an ordinary differential equation in the form…”

Section: Application Of the Laplace Transformationmentioning

confidence: 99%

“…The heat conduction problems in multi-layered bodies based on the Fourier law is the subject of paper [1] by Haji-Sheikh and Beck. Exact solutions of these problems for multi-layered slabs, cylinders and spheres are presented by Özişik in book [2].…”

Section: Introductionmentioning

confidence: 99%

Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab

Kukla¹,

Siedlecka²

2015

J. Appl. Math. Comput. Mech.

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Abstract. In this paper the Laplace transformation for solving the problem of fractional heat conduction in a two-layered slab has been applied. The different orders of Caputo derivative in the time-fractional equation governed the heat transfer in the layers are assumed. The inverse Laplace transform by using a numerical method is determined. The numerical results obtained by using of the eigenfunctions method and by numerically inverting the Laplace transform are compared.

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“…Feng [16] used a method based on modified Green's functions to compute the unsteady heat transfer of a homogeneous or a composite solid body. Haji-Sheikh et al [17] present different types of Green's functions that are solutions of the heat conduction diffusion equation in multi-dimensional, multi-layer bodies for different boundary conditions, calculating eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

Fundamental solutions for transient heat transfer by conduction and convection in an unbounded, half-space, slab and layered media in the frequency domain

Simões

Tadeu

2005

Engineering Analysis with Boundary Elements

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Analytical Green's functions in the frequency domain are presented for the three-dimensional diffusion equation in an unbounded, halfspace, slab and layered media. These proposed expressions take into account the conduction and convection phenomena, assuming that the system is subjected to spatially sinusoidal harmonic heat line sources and do not require any type of discretization of the space domain. The application of time and spatial Fourier transforms along the two horizontal directions allows the solution of the three-dimensional time convection-diffusion equation for a heat point source to be obtained as a summation of one-dimensional responses. The problem is recast in the time domain by means of inverse Fourier transforms using complex frequencies in order to avoid aliasing phenomenon. Further, no restriction is placed on the source time dependence, since the static response is obtained by limiting the frequency to zero and the high frequency contribution to the response is small.The proposed functions have been verified against analytical time domain solutions, known for the case of an unbounded medium, and the Boundary Element Method solutions for the case of the half-space, slab and layered media. q

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Temperature solution in multi-dimensional multi-layer bodies

Cited by 150 publications

References 10 publications

An analytical solution to the problem of time-fractional heat conduction in a composite sphere

An analytical solution to the problem of time-fractional heat conduction in a composite sphere

Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab

Fundamental solutions for transient heat transfer by conduction and convection in an unbounded, half-space, slab and layered media in the frequency domain

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