# Fractional heat conduction in multilayer spherical bodies

### Stanisław Kukla

,### Urszula Siedlecka

Journal of Applied Mathematics and Computational Mechanics |
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FRACTIONAL HEAT CONDUCTION IN MULTILAYER SPHERICAL BODIES

Stanisław Kukla, Urszula Siedlecka

Institute of Mathematics, Czestochowa
University of Technology

Częstochowa, Poland

stanislaw.kukla@im.pcz.pl, urszula.siedlecka@im.pcz.pl

Received: 29
September 2016; accepted: 02 November 2016

**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.

*Keywords: **heat conduction,
multilayer bodies, Caputo derivative, spherical coordinate*

1. Introduction

The heat conduction in multilayer bodies with assumption of the Fourier law of heat transfer has been considered by Özişik in book [1]. Derivations of the temperature distributions in the bodies in rectangular, cylindrical and spherical coordinate systems were presented. 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].

A generalization of the Fourier law leads to a fractional heat conduction equation. The fractional differential equation governing the heat equation includes the fractional derivatives with respect to time and/or space variables. Properties of the fractional derivatives and methods to the solution of the fractional differential equations are presented in books [4-6]. A method of solution of a time-frac-tional heat conduction equation in a solid sphere has been discussed by Ning and Jiang in paper [7]. The time-fractional heat conduction in a multi-layered solid sphere assuming spherical symmetry was the subject of paper [8]. Heat conduction modelling using fractional order derivatives is presented by Žecová and J. Terpák in paper [9].

In this paper, an analytical solution of the time-fractional heat conduction for two-dimensional multilayered spherical bodies is presented. The condition for ensuring the perfect contact at interfaces and the mathematical Robin boundary conditions at boundary surfaces are assumed.

2. Formulation of the problem

Consider *n* spherical concentric layers
which are in perfect thermal contact.
The *i*-th layer () occupied a region described by the spherical coordinates: , (), ,
where is the radial coordinate,
is the polar angle and is the azimuthal angle (Fig. 1).

Fig. 1. A schematic diagram of the *n*-layered
hemisphere

We suppose
that the *i*-th layer is characterized by constant thermal conductivity and constant thermal
diffusivity . Moreover, assuming that the temperature doesn’t depend on
the azimuthal angle, the time fractional heat conduction
in the layers is governed by the following differential equation [10]:

(1) |

where denotes fractional order of the Caputo derivative with respect to time , . The Caputo derivative is defined by [11]

(2) |

In order to simplify the equation (1) we introduce a new variable which is related to the polar angle by

(3) |

Taking into account this relationship in equation (1) we obtain

(4) |

where . The differential equations (4) are complemented by boundary conditions and the conditions providing the perfect thermal contact of the neigh-bouring layers. The mathematical conditions are [2, 10]

(5) |

(6) |

or | (7) |

(8) |

(9) |

where are inner and outer heat transfer coefficients and are inner and outer ambient temperatures. The initial condition is assumed in the form

(10) |

3. Solution of the problem

We search a solution to the problem (1), (4)-(10) in the form of a sum

(11) |

where the function satisfies homogeneous fractional heat conduction differential equation and homogeneous boundary conditions and the function is a solution of a steady-state problem. Substituting the solution (11) into equations (1), (4)-(10) we obtain the problems for the functions and . For we have

(12) |

(13) |

(14) |

or | (15) |

(16) |

(17) |

(18) |

The functions satisfy the following boundary problem

(19) |

(20) |

(21) |

or | (22) |

(23) |

(24) |

3.1. Solution to the homogeneous problem

We find the time dependent function as a solution of the problem (12)-(18), by using the separation of variables method. Substituting the product of functions

(25) |

into equation (12), we obtain three differential equations

(26) |

(27) |

(28) |

where, and are separation constants.

*Solution of the
equation (26)*

Assuming that , the general solution of equation (26) can be written in the form

(29) |

where and are the Legendre functions of order . Because , we assume . Taking into account that

(30) |

and substituting function in the form (25) into equation (15), we obtain an eigenvalue equation

or | (31) |

Solving equation (31a) or (31b), we obtain a sequence of roots . The functions corresponding to the values of create an orthogonal system of functions, i.e. the functions satisfy the orthogonality condition

(32) |

*Solution of the equation (27)*

The general solution of equation (27) has the form

(33) |

where and are spherical Bessel functions of the first and second kind, respectively. Substituting function into equation (25) and next, the obtained function into conditions (13)-(14) and (16)-(17), the system of homogeneous equations is received. We rewrite the equation system in the matrix form

(34) |

where and .

The non-zero solutions of equation (34) exist for these values of for which the determinant of the matrix disappears, i.e.

(35) |

Solving this equation for , the sequences of are obtained. For each value of , the coefficients occurring in equation (33) are determined by solving equation (34). The functions corresponding to the values of satisfy the orthogonality condition

(36) |

*Solution of the equation (28)*

Based on the orthogonality conditions (32), (36) and using (25) in the condition (18), we find the initial condition for the function

(37) |

The solution of the fractional differential equation (28) satisfying the initial condition (37) has the form

(38) |

where is Mittag-Leffler function [12].

Ultimately, using (25), (30), (33) and (38), we have

(39) |

where is given by equation (38).

3.2. Solution to the steady-state problem

We seek a solution of the problem (19)-(24) in the form of a series

(40) |

where are roots of the equation (31a) or (31b). Substituting the function into equation (19) we obtain an Euler differential equation

(41) |

Next, taking into account function (40) in equations (20)-(21) and (23)-(24) and using the orthogonality condition (32), the boundary conditions for the functions are obtained

(42) |

(43) |

(44) |

(45) |

The general solution of the Euler equation (41) has the form

(46)

Substitution (46) in conditions (42)-(45) gives linear non-homogeneous equations which can be written in the matrix form

(47)

where , and . The equation (47) is than solved with respect to unknown . The determined coefficients are then used in equation (46). Thus, the function as a solution of the steady-state problem is given by equation (40) where the functions are defined by equation (46).

Finally, the temperature distribution in the spherical layers received on the basis of the fractional heat conduction model is given by equation (11), (39) and (40).

4. Numerical example

We use the
solution of the heat conduction problem derived in Section 3 to
numerical calculations of the temperature distribution in a layered hemisphere (). We assume that zero temperature is
established at the surface ,
i.e. the boundary conditions (15a) and (22a),
are satisfied. The considered hemisphere consists of five concentric layers
whose locations are determined by
non-dimensional radii: , where . The non-dimensional radii , thermal conductivity and thermal diffusivity in the *i*-th layer of the sphere
are: , , , *i* = 1,…,5. The inner and outer heat
transfer coefficients are: ,
the inner and outer ambient temperatures are: and
the initial temperature is assumed as constant: . The
computations were performed for various values of the order of fractional
time-derivative: . The temperature
distributions on the surfaces of the layers determined for , are shown in Figures 2a-d. The
temperature depends on the order of the
time-derivative occurring in the heat conduction model. This dependence is more
significant for higher temperature of the sphere.

Fig. 2. The non-dimensional temperatures at outer surface and at interfaces of the layered hemisphere: a) , b) , c) , d)

5. Conclusions

*An
analytical *solution of
the problem of time-fractional heat conduction in
two-dimensional *multi-layer spherical bodies has been presented. The
formulation of the problem includes the heat conduction in the spherical bodies
which occupied regions defined by finite intervals of the radial coordinate and
polar angle.
*The conditions to perfect contact at
interfaces and the mathematical Robin boundary conditions were assumed. The
derived solution applies to the bodies consisting of an arbitrary number of
layers which are characterized by different thermal
conductivity and thermal diffusivity. The numerical example shows that the
order of fractional time-derivative is of significant importance for
temperature distribution in the body. The temperature at the outer surface and at
interfaces of
the layered hemisphere is lower order of the fractional time-derivative. The
further research will take into consideration the fractional heat conduction in
spherical multilayer bodies with time-dependent boundary conditions.

References

[1] Özişik M.N., Heat conduction, Wiley, New York 1993.

[2] Lu X., Viljanen M., An analytical method to solve heat conduction in layered spheres with time-dependent boundary conditions, Physics Letters A 2006, 351, 274-282.

[3] Jain P.K., Singh S., Rizwan-uddin, An exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical coordinates, International Journal of Heat and Mass Transfer 2010, 53, 2133-2142.

[4] Diethelm K., The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin Heidelberg 2010.

[5] Podlubny I., Fractional Differential Equations, Academic Press, San Diego 1999.

[6] Povstenko Y., Linear Fractional Diffusion-wave Equation for Scientists and Engineers, Birkhäuser, New York 2015.

[7] Ning T.H., Jiang X.Y., Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation, Acta Mechanica Sinica 2011, 27(6), 994-1000.

[8] Kukla S., Siedlecka U., An analytical solution to the problem of time-fractional heat conduction in a composite sphere, Bulletin of the Polish Academy of Sciences, Technical Sciences (in print).

[9] Žecová M., Terpák J., Heat conduction modeling by using fractional-order derivatives, Applied Mathematics and Computation 2015, 257, 365-373.

[10] Povstenko Y., Fractional heat conduction in an infinite medium with a spherical inclusion, Entropy 2013, 15, 4122-4133.

[11] Ishteva M., Scherer R., Boyadjiev L., On the Caputo operator of fractional calculus and C-Laguerre functions, Mathematical Sciences Research Journal 2005, 9(6), 161-170.

[12] Haubold H.J., Mathai A.M., Saxena R.K., Mittag-Leffler functions and their applications, Journal of Applied Mathematics, Article ID 298628, 2011.