Transient analytical solution to heat conduction in composite circular cylinder

Lü, Xiaoshu; Tervola, Pekka; Viljanen, Martti

doi:10.1016/j.ijheatmasstransfer.2005.06.019

Cited by 77 publications

(42 citation statements)

References 8 publications

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“…, (2) where m ¡ 1 < α < m, m 2 N = {1, 2, …}. The geometric and physical interpretation of the fractional derivatives are given in [28].…”

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

confidence: 99%

“…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%

“…wherein the term tg i ¤ (r) under the second integral occurs only for α 2 (1,2]. In (31), E α, β is a two-parameter Mittag-Leffler function, which is defined by [29]:…”

Section: Solution Of the Problemmentioning

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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“…, (2) where m ¡ 1 < α < m, m 2 N = {1, 2, …}. The geometric and physical interpretation of the fractional derivatives are given in [28].…”

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

confidence: 99%

“…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%

“…wherein the term tg i ¤ (r) under the second integral occurs only for α 2 (1,2]. In (31), E α, β is a two-parameter Mittag-Leffler function, which is defined by [29]:…”

Section: Solution Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Application of the Green's function to such problems can be found in the book by Beck et al [1], in the book by Özişik [2] and in the book by Duffy [3], as well as in several papers, for example in references [4][5][6][7]. The advantage of the Green's function method lies in the fact that the solution of the non-homogeneous problems can be expressed in terms of the Green's function.…”

Section: Introductionmentioning

confidence: 99%

Green's function for heat conduction problems in a multi-layered hollow cylinder

Kukla¹,

Siedlecka²

2014

J. Appl. Math. Comput. Mech.

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Abstract. In this paper, derivation of the Green's function for the heat conduction problems in a finite multi-layered hollow cylinder is presented. Formulation and solution of the problem includes an arbitrary number of the cylinder layers characterized by various thermal properties. At the interfaces perfect thermal contact was assumed. The Green's function for the three-dimensional heat conduction problems in the cylindrical coordinate has been presented in the form of a product of two other Green's functions.

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Numerical Analysis of Thermal Development of Laminar Flow in a Concentric Multilayer Annulus

et al. 2013

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The thermal behavior of laminar flow in a concentric multilayer annulus is investigated numerically for varying dimensionless inner radii and a defined number of fluid layers. Under a hydrodynamically developed flow assumption within the fluid layers, the development of temperature profiles and heat transfer along the annulus are analyzed for two different boundary conditions. The mean temperature distribution, local Nusselt number, and mean Nusselt number are discussed in detail with an emphasis on the effects of the inner radius and number of fluid layers. The obtained results indicate that the change in heat transfer coefficient in a layered annulus is more pronounced at a small inner radius or larger radius ratio. A further insertion of more than ten layers has insignificant influence on the convective heat transfer in a layered annulus.

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Transient analytical solution to heat conduction in composite circular cylinder

Cited by 77 publications

References 8 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

Green's function for heat conduction problems in a multi-layered hollow cylinder

Numerical Analysis of Thermal Development of Laminar Flow in a Concentric Multilayer Annulus

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