An exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical coordinates

Jain, Prashant; Singh, Suneet

doi:10.1016/j.ijheatmasstransfer.2009.12.035

Cited by 64 publications

(25 citation statements)

References 24 publications

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“…Heat conduction in layered bodies in spherical coordinates was recently investigated in [4][5][6][7][8][9][10]. An analytical solution to the problem of heat conduction in a multilayered sphere with time-dependent boundary conditions was derived by Lu and Viljanen in [4].…”

Section: Introductionmentioning

confidence: 99%

“…The solution was obtained using the Green's function method. An analytical series solution for a two-dimensional, transient boundary-value problem for multilayered heat conduction in spherical coordinates has been presented by Jain et al in [6]. In the formulation of the problem, time-independent volumetric heat sources in the concentric layers were assumed.…”

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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Section: Introductionmentioning

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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“…The heat conduction in layered spheres with time-dependent boundary conditions assuming spherical symmetry was the subject of paper [2]. A solution of the heat conduction problem for a two-dimensional multi-layered sphere, hemisphere, spherical cone and spherical wedge presents paper [3].…”

Section: Introductionmentioning

confidence: 99%

Fractional heat conduction in multilayer spherical bodies

Kukla

Siedlecka

2016

J. Appl. Math. Comput. Mech.

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Abstract. In this paper an analytical solution of the time-fractional heat conduction problem in a spherical coordinate system is presented. The considerations deal the two--dimensional problem in multilayer spherical bodies including a hollow sphere, hemisphere and spherical wedge. The mathematical Robin conditions are assumed. The solution is a sum of time-dependent function satisfied homogenous boundary conditions and of a solution of the steady-state problem. Numerical example shows the temperature distributions in the hemisphere for various order of time-derivative.

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“…Transient heat transfer in multilayered composite plates, cylinders and spheres are of great interest in a number of engineering applications [4,5,6,7,8,9,10,11,12]. The use of composite media is necessary when the thermal and mechanical properties of a single layer is not sufficient as to fulfill both thermal and mechanical requirements.…”

Section: Introductionmentioning

confidence: 99%

Lumped models for transient thermal analysis of multilayered composite pipeline with active heating

2015

Applied Thermal Engineering

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An exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical coordinates

Cited by 64 publications

References 24 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

Fractional heat conduction in multilayer spherical bodies

Lumped models for transient thermal analysis of multilayered composite pipeline with active heating

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