A Temperature Total Fourier Series Solution For a Hollow Sphere

In the following pages, we exhibit an analytical solution of a two-dimensional temperature field in a hollow sphere under total periodic boundary condition. The material is assumed to be homogeneous and isotropic with time-independent thermal properties .Till now periodic boundary condition was derived with a harmonic vibration, whereas there is a noticeable difference in the practical conditions with harmonic vibration .In this essay, by means of Fourier analysis, we imagine the outside total periodic boundary condition, as aggregate of harmonic vibrations .To solve the problem, first we imagine the boundary condition as constant values and with separation of variables; we can obtain temperature distribution in the sphere. Then Duhamel’s theorem is used to calculate temperature field under fully periodic boundary condition. For confirmation of accurate solution, we can compare the result for a harmonic vibration and those reported by others. Also, solutions for a hollow sphere were compared with other present references. At last we can obtain thermal stresses which is caused by temperature field in the hollow sphere Keywords— Hollow Sphere, Fourier Series , conduction


I INTRODUCTION
There are several heat conduction problems that can be modeled by a sphere of constant properties, for example, food freezing and the hydrocooling of spherical fruits or vegetables [1]. The solution of some cases of heat conduction problems can be found in heat transfer literature. Heat conduction problems with periodic boundary condition have some applications in engineering such as penetration of the daily and annual temperature cycles into the earth's surface, heating up and cooling down phases in the Siemens Martin glass melting furnaces, wall temperature oscillation of internal combustion engines, experimental methods for specifying the thermal diffusivity of materials [2] and the temperature field [3,4], and also the thermal stresses caused by temperature distribution. Trostel calculated thermal stresses caused by thermal loads in a solid sphere [5]. Zubair and Chaudhry discussed the solution for temperature and heat flux in a semi-infinite solid subject to periodic-type surface heat fluxes [6]. The calculation of temperature distribution in a solid sphere under a periodic boundary condition is presented in [4] that is simulated by harmonic oscillation of the ambient temperature. The purpose of this paper is to derive an analytical solution for a two-dimensional heat conduction in a hollow sphere, subjected to a periodic boundary condition. As for the validity of the results, a comparison between the temperature distribution in a solid sphere with the theoretical ones [4] is presented for the same boundary condition. The results can be used for approximation to the real problems with periodic boundary condition. They can also be utilized to verify the time consuming complex computer calculations.

II MATHEMATICAL MODEL
The heat conduction equation in spherical coordinates for an isotropic material that has temperature and timeindependent properties, and without heat source under axisymmetric condition, is ,: The initial temperature of the ambient and the hollow sphere are zero. The inner boundary condition is insulated and the outer one is assumed to be boundary condition of type 3: We consider that where ) (t g is assumed to be a periodic function that is decomposed using Fourier series: An easy way to comply with the conference paper formatting requirements is to use this document as a template and simply type your text into it. and ) (ψ o f is an arbitrary function.

A. Analytical Solution
The problem cannot be solved directly because of the dependency of nonhomogeneous term, ) . The solution of a heat-conduction problem with timedependent boundary condition can be related to the solution of the same problem with time-independent boundary condition by means of Duhamel's theorem. Thus, first of all, the equation should be solved with the assumption that the boundary condition is time independent. In this situation the boundary and initial conditions are: In this case, use was made of the superposition principle; the solution is the sum of a steady solution, where the boundary condition is given by Eq. (4). The transient differential equation is: and the following conditions must be satisfied To solve Eq. (6), the method of separation of variables is used. Two differential equations are obtained, a Euler type and a Legendre type. Then, by applying Eq. (4), the solution of the steady state is:

D. Temperature Field Under Time Varying Boundary
Condition.
As we mentioned before, the temperature distribution under a constant boundary condition is the summation of steady and transient states: Equation (17) It can be assumed that τ d dC n are constant at some time, τ Therefore the temperature distribution after τ − t seconds after the beginning of the influences can be expressed in the form (7): o n a n n kn n n kn n r n k kn dC d r t r e r r r r dr P d d Thus, the temperature field can be obtained by summation of all exponential terms should vanish, hence ) (t T kn takes the form: Thus, the temperature distribution in a hollow sphere for a periodic boundary condition when the inner boundary is insulated and the outer one dissipates heat into the ambient can be expressed in the form: Where m A is the ratio of the oscillation amplitude of temperature distribution in the sphere and the ambient temperature with the same frequency and kn φ is the phase difference. By calculating and plotting Eq. (35), the maximum amplitude of summation of harmonic waves and the phase difference can be obtained.

III Results and Discussion
To check our series solution, as a special case, we solved the problem of a solid sphere with a harmonic boundary condition with our method and compared the results with the corresponding ones in the literature [5]. As presented in Figs Figures 3-8 show the oscillation amplitude and the phase difference for the hollow sphere under a periodic boundary condition, which is decomposed using Fourier series with different radii and polar angles. Fig. 1 Comparison between the result of the amplitude of a one-dimensional temperature field of a solid sphere [8] and the results for a solid sphere presented in [4] under the same boundary condition Fig. 2 Comparison between the result of the phase difference of a onedimensional temperature field of a solid sphere [8] and the results for a solid sphere presented in [4] under the same boundary condition