research article exact analytical solution for 3d time ...solutions for d multilayer transient heat...

Research ArticleExact Analytical Solution for 3D Time-Dependent HeatConduction in a Multilayer Sphere with Heat Sources UsingEigenfunction Expansion Method

Nemat Dalir

Department of Mechanical Engineering, Salmas Branch, Islamic Azad University, Salmas, Iran

Correspondence should be addressed to Nemat Dalir; [email protected]

Received 4 June 2014; Accepted 30 September 2014; Published 18 November 2014

Academic Editor: Jose C. Merchuk

Copyright © 2014 Nemat Dalir. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An exact analytical solution is obtained for the problem of three-dimensional transient heat conduction in the multilayered sphere.The sphere has multiple layers in the radial direction and, in each layer, time-dependent and spatially nonuniform volumetricinternal heat sources are considered. To obtain the temperature distribution, the eigenfunction expansion method is used. Anarbitrary combination of homogenous boundary condition of the first or second kind can be applied in the angular and azimuthaldirections. Nevertheless, solution is valid for nonhomogeneous boundary conditions of the third kind (convection) in the radialdirection. A case study problem for the three-layer quarter-spherical region is solved and the results are discussed.

1. Introduction

Multilayer materials are composite media composed of sev-eral layers. Because of the additional benefit of combiningvarious mechanical, physical, and thermal properties of dif-ferent substances, a construction using multilayer elementsis of interest. Multilayer materials are used in semicircularfiber insulated heaters, multilayer insulation materials, andnuclear fuel rods. Multilayer transient heat conduction findsapplications in thermodynamics, fuel cells, and electrochem-ical reactors. The layered sphere is utilized to investigate thethermal properties of composite media by assuming embed-ded spherical particles in the composite matrix. For solvingthe problems of multilayer transient heat conduction, thesame methods which are used in solving problems of singlelayer transient heat conduction are applied. These methodscan be classified into two groups: analytical methods andnumerical methods. Analytical methods are advantageousover numerical methods in two ways: (1) analytical solutionscan be used as benchmark to examine and actually confirmnumerical algorithms; (2) compared to a discrete numericalsolution, the mathematical form of an analytical solutioncan provide better insight. It should also be mentionedthat the analytical methods applied to multilayer transient

conduction are analogous to those used in the single-layertransient heat conduction. These analytical methods includeGreen’s function method, the Laplace transform, separationof variables, and eigenfunction expansion method.

Many researchers have solved the transient heat con-duction problem in a composite medium. For instance, Salt[1] solved the transient heat conduction problem in a two-dimensional composite slab using an orthogonal eigenfunc-tion expansion technique. Mikhailov and Özişik [2], usingthe orthogonal expansion approach, solved the problem oftransient three-dimensional heat conduction in a compositeCartesian medium. Haji-Sheikh and Beck [3] used Green’sfunction method to obtain temperature distribution in athree-dimensional two-layer orthotropic slab. deMonte [4, 5]applied the eigenfunction expansion method to obtain thetransient temperature distribution for the heat conduction ina two-dimensional two-layer isotropic slab with homogenousboundary conditions. Lu et al. [6] and Lu and Viljanen [7]combined separation of variables and Laplace transformsto solve the transient conduction in the two-dimensionalcylindrical and spherical media. Singh et al. [8, 9] andJain et al. [10, 11] used the combination of separation ofvariables and eigenfunction expansion methods to solve thetwo-dimensional multilayer transient heat conduction in

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014, Article ID 708723, 8 pageshttp://dx.doi.org/10.1155/2014/708723

2 International Scholarly Research Notices

spherical coordinates. Recently, Dalir andNourazar [12] usedthe eigenfunction expansion method to solve the problem ofthree-dimensional transient heat conduction in a multilayercylinder.

Singh et al. [8, 9] and Jain et al. [10, 11] have studied2D multilayer transient conduction problems in sphericaland cylindrical coordinates. They have obtained analyticalsolutions for 2D multilayer transient heat conduction inspherical coordinates, in polar coordinates with multiplelayers in the radial direction, and in a multilayer annulus.They have used the method of partial solutions to obtainthe temperature distributions. In the method of partialsolutions, the nonhomogeneous transient problem is splitinto two subproblems: a nonhomogeneous steady-state sub-problem and a homogeneous transient subproblem. Then,the eigenfunction expansion method is used to solve thenonhomogeneous steady-state subproblem and the methodof separation of variables is used to solve the homogeneoustransient subproblem.

The literature survey for the exact analytical solution for3D transient heat conduction in multilayered sphere demon-strates that such a solution has not, so far, been developed.Thus, in the present paper, using the eigenfunction expansionmethod, an analytical triple-series solution for transientheat conduction in the 3D spherical coordinates for radialmultilayer domain with spatially nonuniform and time-dependent internal heat sources is obtained. Homogenousboundary conditions of the first or second kind can be appliedon surfaces of 𝜃 = constant and 𝜙 = constant. However,nonhomogeneous boundary conditions of the third kind(convection) [11] are used in the 𝑟-direction.

Some assumptions are made for the 3D multilayer spher-ical transient conduction problem. First, the problem is aboundary-value problem of conduction in spherical (𝑟-𝜃-𝜙coordinates) or part-sphericalmultilayer geometries. Second,volumetric internal heat sources of nonuniform and time-dependent (𝑟, 𝜃, 𝜙 and 𝑡-dependent) types are present. Third,on the inner and outer radial boundaries, nonhomogeneousboundary conditions of any kind can be used but, on theboundary surfaces in the 𝜃 and 𝜙-directions, only the firstor second kind of homogeneous boundary condition can beapplied.

2. Mathematical Formulation

A 𝑛-layer composite spherical slab (𝑟0≤ 𝑟 ≤ 𝑟

𝑛, 0 ≤ 𝜃 ≤

𝜓, and 0 ≤ 𝜙 ≤ 𝜔) is considered. All the layers have perfectthermal contact and are presumed to be isotropic in thermalproperties. 𝛼

𝑖and 𝑘

𝑖are the temperature independent ther-

mal diffusivity and thermal conductivity of the 𝑖th layer. At𝑡 = 0, the 𝑖th layer is at a specified temperature 𝑓

𝑖(𝑟, 𝜃, 𝜙)

and time dependent heat sources 𝑔𝑖(𝑟, 𝜃, 𝜙, 𝑡) are switched

on in each radial layer. For 𝑡 > 0, homogenous boundaryconditions of first or second kind are applied to the angularsurfaces of 𝜃 = 0 and 𝜃 = 𝜓 and azimuthal surfaces of 𝜙 = 0and 𝜙 = 𝜔. For the inner (𝑖 = 1, 𝑟 = 𝑟

0) and the outer

(𝑖 = 𝑛, 𝑟 = 𝑟𝑛) radial surfaces, all three kinds of boundary

conditions are applicable.

The governing differential equation of the 3D transientconduction in a multilayered sphere with heat sources is asfollows:

𝜕2𝑇𝑖

𝜕𝑟2+2

𝑟

𝜕𝑇𝑖

𝜕𝑟+1

𝑟2𝜕2𝑇𝑖

𝜕𝜃2+cot 𝜃𝑟2

𝜕𝑇𝑖

𝜕𝜃+

1

𝑟2sin2𝜃𝜕2𝑇𝑖

𝜕𝜙2

+𝑔𝑖(𝑟, 𝜃, 𝜙, 𝑡)

𝑘𝑖

=1

𝛼𝑖

𝜕𝑇𝑖

𝜕𝑡,

𝑇𝑖= 𝑇𝑖(𝑟, 𝜃, 𝜙, 𝑡) , 𝑟

0≤ 𝑟 ≤ 𝑟

𝑛,

𝑟𝑖−1≤ 𝑟 ≤ 𝑟

𝑖, 1 ≤ 𝑖 ≤ 𝑛,

0 ≤ 𝜃 ≤ 𝜓, 𝜓 < 𝜋,

0 ≤ 𝜙 ≤ 𝜔, 𝜔 < 2𝜋,

𝑡 ≥ 0.

(1)

The boundary conditions are as follows.

(i) Inner surface of 1st layer (𝑖 = 1):

𝐴 in𝜕𝑇1

𝜕𝑟(𝑟0, 𝜃, 𝜙, 𝑡) + 𝐵in𝑇1 (𝑟0, 𝜃, 𝜙, 𝑡) = 𝐶in. (2)

(ii) Outer surface of 𝑛th layer (𝑖 = 𝑛):

𝐴out𝜕𝑇𝑛

𝜕𝑟(𝑟𝑛, 𝜃, 𝜙, 𝑡) + 𝐵out𝑇𝑛 (𝑟𝑛, 𝜃, 𝜙, 𝑡) = 𝐶out. (3)

(iii) 𝜃 = 𝜓 surface (𝑖 = 1, 2, . . . , 𝑛):

𝑇𝑖(𝑟, 𝜃 = 𝜓, 𝜙, 𝑡) = 0 or

𝜕𝑇𝑖

𝜕𝜃(𝑟, 𝜃 = 𝜓, 𝜙, 𝑡) = 0,

𝑖 = 1, . . . , 𝑛.

(4)

(iv) 𝜙 = 0 surface (𝑖 = 1, 2, . . . , 𝑛):

𝑇𝑖(𝑟, 𝜃, 𝜙 = 0, 𝑡) = 0 or

𝜕𝑇𝑖

𝜕𝜃(𝑟, 𝜃, 𝜙 = 0, 𝑡) = 0,

𝑖 = 1, . . . , 𝑛.

(5)

(v) 𝜙 = 𝜔 surface (𝑖 = 1, 2, . . . , 𝑛):

𝑇𝑖(𝑟, 𝜃, 𝜙 = 𝜔, 𝑡) = 0 or

𝜕𝑇𝑖

𝜕𝜃(𝑟, 𝜃, 𝜙 = 𝜔, 𝑡) = 0,

𝑖 = 1, . . . , 𝑛.

(6)

(vi) Inner interface of the 𝑖th layer (𝑖 = 2, . . . , 𝑛):

𝑇𝑖(𝑟𝑖−1, 𝜃, 𝜙, 𝑡) = 𝑇

𝑖−1(𝑟𝑖−1, 𝜃, 𝜙, 𝑡) 𝑖 = 2, . . . , 𝑛,

𝑘𝑖

𝜕𝑇𝑖

𝜕𝑟(𝑟𝑖−1, 𝜃, 𝜙, 𝑡) = 𝑘

𝑖−1

𝜕𝑇𝑖−1

𝜕𝑟(𝑟𝑖−1, 𝜃, 𝜙, 𝑡) 𝑖 = 2, . . . , 𝑛.

(7)

International Scholarly Research Notices 3

(vii) Outer interface of the 𝑖th layer (𝑖 = 1, . . . , 𝑛 − 1):

𝑇𝑖(𝑟𝑖, 𝜃, 𝜙, 𝑡) = 𝑇

𝑖+1(𝑟𝑖, 𝜃, 𝜙, 𝑡) 𝑖 = 1, . . . , 𝑛 − 1, (8)

𝑘𝑖

𝜕𝑇𝑖

𝜕𝑟(𝑟𝑖, 𝜃, 𝜙, 𝑡) = 𝑘

𝑖+1

𝜕𝑇𝑖+1

𝜕𝑟(𝑟𝑖, 𝜃, 𝜙, 𝑡) 𝑖 = 1, . . . , 𝑛 − 1.

(9)

The initial condition is as follows:

𝑇𝑖(𝑟, 𝜃, 𝜙, 𝑡 = 0) = 𝑓

𝑖(𝑟, 𝜃, 𝜙) , 1 ≤ 𝑖 ≤ 𝑛. (10)

It is worth mentioning that, at 𝑟 = 𝑟0and 𝑟 = 𝑟

𝑛, boundary

conditions of first, second, or third kind are applied byappropriate selection of the coefficients in (2) and (3). Itshould also be mentioned that zero inner radius (𝑟

0= 0) for

multilayered sphere ismodeled by assigning zero values to𝐵inand 𝐶in in (2) [8].

3. Solution Methodology

The eigenfunction expansion method is used to solve theproblem. In the eigenfunction expansion method, first, byusing the associated eigenvalue problem (∇2𝜑 = −𝜆2𝜑),the eigenfunctions are attained at every spatial direction ofthe problem. The associated eigenvalue problem is solved bythe use of separation of variables. Afterward, the dependentvariable and the available nonhomogeneity in the governingdifferential equation of the problem are separately written asseries expansions of the eigenfunctions. In heat conductionproblems, the dependent variable is temperature and theavailable nonhomogeneity is the volumetric heat source. Theseries expansions are then substituted into the differentialequation. By performing some mathematical manipulations,an ordinary differential equation (ODE) is finally obtainedfor the independent variable. The solution of the problem iscompleted by solving this ODE, which is a first order ODE inthe case of heat conduction problems.

As stated before, the method of partial solutions wasused by Jain and Singh [11] for solving 2D transient heatconduction problems, the reason being that the heat sourceis independent of time. However, the method of partialsolutions cannot be used for solving the present 3D transientheat conduction problem because the heat source dependson time. Thus, due to time dependence of the heat source,the partial solution of the steady-state subproblem cannotinclude the heat source term and the partial solutionsmethodcannot be used. Therefore, to the best knowledge of theauthors, the most efficient tool for solving the 3D heatconduction problem of the present paper is the eigenfunctionexpansion method.

For the transient problem of present paper, the associatedeigenvalue problem is written as follows:

∇2𝜑𝑖= −𝜆2𝜑𝑖⇒

𝜕2𝜑𝑖

𝜕𝑟2+2

𝑟

𝜕𝜑𝑖

𝜕𝑟+1

𝑟2𝜕2𝜑𝑖

𝜕𝜃2

+cot 𝜃𝑟2

𝜕𝜑𝑖

𝜕𝜃+

1

𝑟2sin2𝜃𝜕2𝜑𝑖

𝜕𝜙2

= −𝜆2𝜑𝑖.

(11)

Using the method of separation of variables, (11) is solved asfollows:

𝜑𝑖(𝑟, 𝜃, 𝜙) = 𝑅

𝑖(𝑟) Θ𝑖(𝜃)Φ𝑖(𝜙) , (12)

𝑅𝑖Θ𝑖Φ𝑖


+2

𝑟



+1

𝑟2



+cot 𝜃𝑟2



+1

𝑟2sin2𝜃𝑅𝑖Θ𝑖Φ𝑖


= −𝜆2𝑅𝑖Θ𝑖Φ𝑖


,

(13)

𝑅𝑖

𝑅𝑖

+2

𝑟

𝑅𝑖

𝑅𝑖

+1

𝑟2

Θ𝑖

Θ𝑖

+cot 𝜃𝑟2

Θ𝑖

Θ𝑖

+1

𝑟2sin2𝜃Φ𝑖

Φ𝑖

= −𝜆2, (14)

𝑅𝑖

𝑅𝑖

+2

𝑟

𝑅𝑖

𝑅𝑖

+1

𝑟2

Θ𝑖

Θ𝑖

+cot 𝜃𝑟2

Θ𝑖

Θ𝑖

= −1

𝑟2sin2𝜃Φ𝑖

Φ𝑖

− 𝜆2, (15)

sin2𝜃(𝑟2𝑅𝑖

𝑅𝑖

+ 2𝑟𝑅𝑖

𝑅𝑖

+Θ𝑖

Θ𝑖

+ cot 𝜃Θ𝑖

Θ𝑖

+ 𝜆2𝑟2)

= −Φ𝑖

Φ𝑖

− = +]2,

(16)

Φ

𝑖+ ]2𝑖𝑚Φ𝑖= 0 → Φ

𝑖𝑚(𝜙) = 𝑐

1sin ]𝑖𝑚𝜙 + 𝑐2cos ]𝑖𝑚𝜙,

𝑟2𝑅𝑖

𝑅𝑖

+ 2𝑟𝑅𝑖

𝑅𝑖

+Θ𝑖

Θ𝑖

+ cot 𝜃Θ𝑖

Θ𝑖

+ 𝜆2𝑟2=

]2

sin2𝜃,

(17)

→ 𝑟2𝑅𝑖

𝑅𝑖

+ 2𝑟𝑅𝑖

𝑅𝑖

+ 𝜆2𝑟2= −

Θ𝑖

Θ𝑖

− cot 𝜃Θ𝑖

Θ𝑖

+]2

sin2𝜃

= +𝛽2,

(18)

𝑟2𝑅

𝑖+ 2𝑟𝑅

𝑖+ (𝜆2

𝑖𝑝𝑛𝑟2− 𝛽2

𝑛) 𝑅𝑖= 0

→ 𝑅𝑖𝑝𝑛(𝑟) =

1

√𝑟[𝑐3𝐽𝛽𝑛+0.5

(𝜆𝑖𝑝𝑛𝑟)

+ 𝑐4𝑌𝛽𝑛+0.5

(𝜆𝑖𝑝𝑛𝑟)] ,

Θ

𝑖+ (cot 𝜃)Θ

𝑖+ (𝛽2−

]2

sin2𝜃)Θ𝑖= 0.

(19)

By using the following change of variable:

𝜇 = cos 𝜃 → 1 − 𝜇2 = sin2𝜃 (20)the first and second derivatives of Θ with respect to 𝜃 areobtained as follows:

Θ=𝑑Θ𝑖

𝑑𝜃=𝑑Θ𝑖

𝑑𝜇

𝑑𝜇

𝑑𝜃= − sin 𝜃

𝑑Θ𝑖

𝑑𝜇,

Θ

𝑖=𝑑2Θ𝑖

𝑑𝜃2=𝑑

𝑑𝜃(𝑑Θ𝑖

𝑑𝜃) =

𝑑

𝑑𝜃(− sin 𝜃

𝑑Θ𝑖

𝑑𝜇)

= − cos 𝜃𝑑Θ𝑖

𝑑𝜇− sin 𝜃 𝑑

𝑑𝜃(𝑑Θ𝑖

𝑑𝜇)

= sin2𝜃𝑑2Θ𝑖

𝑑𝜇2− cos 𝜃

𝑑Θ𝑖

𝑑𝜇.

(21)


Substituting (21) in (19) results in the following:

Θ

𝑖+ (cot 𝜃)Θ

𝑖+ (𝛽2−

]2

sin2𝜃)Θ𝑖= 0

→ sin2𝜃𝑑2Θ𝑖

𝑑𝜇2− cos 𝜃

𝑑Θ𝑖

𝑑𝜇

+ (cos 𝜃sin 𝜃

)(− sin 𝜃𝑑Θ𝑖

𝑑𝜇) + (𝛽

2−

]2

sin2𝜃)Θ𝑖= 0

→ (1 − 𝜇2)𝑑2Θ𝑖

𝑑𝜇2− 2𝜇

𝑑Θ𝑖

𝑑𝜇+ (𝛽2

𝑛−

]2𝑖𝑚

1 − 𝜇2)Θ𝑖= 0.

(22)

If 𝛽2𝑛= 𝑛(𝑛 + 1), (22) is the associated Legendre equation. Its

solution is written as follows:

Θ𝑖𝑛(𝜇) = 𝑐

5𝑃]𝑖𝑚

𝑖𝑛(𝜇) + 𝑐

6𝑄

]𝑖𝑚

𝑖𝑛(𝜇)

→ Θ𝑖𝑛 (𝜃) = 𝑐5𝑃

]𝑖𝑚

𝑖𝑛(cos 𝜃) + 𝑐6𝑄

]𝑖𝑚

𝑖𝑛(cos 𝜃)

𝑄]𝑖𝑚

𝑖𝑛(cos 0) =𝑄]𝑖𝑚

𝑖𝑛(1) =∞→𝑐

6= 0

→ Θ𝑖𝑛(𝜃) = 𝑃

]𝑖𝑚

𝑖𝑛(cos 𝜃) .

(23)

The problem eigenvalues are 𝜁2𝑖𝑚𝑛𝑝

= 𝜆2𝑖𝑝𝑛+ ]2𝑖𝑚. It should be

stated that the heat fluxes continuity at the interfaces of theradial layers gives the following:

Φ𝑖𝑚= Φ𝑚, ]

𝑖𝑚= ]𝑚,

𝜆𝑖𝑝𝑛= 𝜆1𝑝𝑛√

𝛼1

𝛼𝑖

.

(24)

The eigenfunctions 𝑅𝑖𝑝𝑛(𝑟), Θ

𝑖𝑛(𝜃), and Φ

𝑖𝑚(𝜙) in 𝑟-, 𝜃-, and

𝜙-directions are derived by applying the boundary conditionsin each direction in the following equation:

𝑅𝑖𝑝𝑛(𝑟) =

1

√𝑟[𝑐3𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟) + 𝑐4𝑌𝑛(𝑛 + 1) + 0.5


Θ𝑖𝑛(𝜃) = 𝑃

]𝑚

𝑖𝑛(cos 𝜃) ,

Φ𝑚(𝜙) = 𝑐

1sin ]𝑚𝜙 + 𝑐2cos ]𝑚𝜙.

(25)

It is assumed that the solution of the problem is in the formof a triple-series expansion of the derived eigenfunctions asfollows:

𝑇𝑖(𝑟, 𝜃, 𝜙, 𝑡) =

∞

∑𝑚=1

∞

∑𝑛 = 1

∞

∑𝑝= 1

𝑇𝑖𝑚𝑛𝑝 (𝑡) 𝑅𝑖𝑝𝑛 (𝑟) Θ𝑖𝑛 (𝜃)Φ𝑚 (𝜙) .

(26)The heat source term is also written as a triple-series expan-sion of the eigenfunctions such that

𝑔𝑖(𝑟, 𝜃, 𝜙, 𝑡) =

∞

∑𝑚=1

∞

∑𝑛 = 1

∞

∑𝑝= 1

𝑔𝑖𝑚𝑛𝑝

(𝑡) 𝑅𝑖𝑝𝑛(𝑟) Θ𝑖𝑛(𝜃)Φ𝑚(𝜙) ,

(27)

where the coefficient 𝑔𝑖𝑚𝑛𝑝

(𝑡) is obtained by the use of theorthogonality property as follows:


(𝑡) = (∫𝜔

0

∫𝜓

0

∫𝑟𝑖

𝑟𝑖 − 1

𝑔𝑖(𝑟, 𝜃, 𝜙, 𝑡) 𝑟

2

× 𝑅𝑖𝑝𝑛(𝑟) Θ𝑖𝑛(𝜃)Φ𝑚(𝜙) 𝑑𝑟 𝑑𝜃 𝑑𝜙)

× (∫𝜔

0

∫𝜓

0

∫𝑟𝑖

𝑟𝑖 − 1

𝑟2𝑅2

𝑖𝑝𝑛(𝑟) Θ2

𝑖𝑛(𝜃)

× Φ2

𝑚(𝜙) 𝑑𝑟 𝑑𝜃 𝑑𝜙)

−1

.

(28)

Substitution of (26) and (27) in (1) results in the following:

𝑇𝑖𝑚𝑛𝑝

(𝑡) 𝑅

𝑖𝑝𝑛(𝑟) Θ𝑖𝑛(𝜃)Φ𝑖𝑚(𝜙) +

2

𝑟𝑇𝑖𝑚𝑛𝑝

(𝑡) 𝑅

𝑖𝑝𝑛(𝑟) Θ𝑖𝑛(𝜃)Φ𝑖𝑚(𝜙) +

1

𝑟2𝑇𝑖𝑚𝑛𝑝

(𝑡) 𝑅𝑖𝑝𝑛(𝑟) Θ

𝑖𝑛(𝜃)Φ𝑖𝑚(𝜙)

+cot 𝜃𝑟2


(𝑡) 𝑅𝑖𝑝𝑛(𝑟) Θ

𝑖𝑛(𝜃)Φ𝑖𝑚(𝜙) +

1

𝑟2sin2𝜃𝑇𝑖𝑚𝑛𝑝

(𝑡) 𝑅𝑖𝑝𝑛(𝑟) Θ𝑖𝑛(𝜃)Φ

𝑖𝑚(𝜙)

+1

𝑘𝑖


(𝑡) 𝑅𝑖𝑝𝑛(𝑟) Θ𝑖𝑛(𝜃)Φ𝑖𝑚(𝜙) =

1

𝛼𝑖

𝑇

𝑖𝑚𝑛𝑝(𝑡) 𝑅𝑖𝑝𝑛(𝑟) Θ𝑖𝑛(𝜃)Φ𝑖𝑚(𝜙) ,

(29)

→𝑑𝑇𝑖𝑚𝑛𝑝 (𝑡)

𝑑𝑡+ (−𝛼

𝑖)(

𝑅𝑖𝑝𝑛(𝑟)

𝑅𝑖𝑝𝑛 (𝑟)

+2

𝑟



+1

𝑟2

Θ𝑖𝑛(𝜃)

Θ𝑖𝑛 (𝜃)

+cot 𝜃𝑟2

Θ𝑖𝑛(𝜃)

Θ𝑖𝑛 (𝜃)

+1

𝑟2sin2𝜃Φ𝑖𝑚(𝜙)

Φ𝑖𝑚(𝜙)

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

= Γ𝑖𝑚𝑛𝑝


(𝑡) =𝛼𝑖

𝑘𝑖


(𝑡) . (30)


Equation (30) is a first-order nonhomogeneous ODE and hasthe following solution:


(𝑡) =𝛼𝑖

𝑘𝑖

𝑒−Γ𝑖𝑚𝑛𝑝𝑡∫𝜏 = 𝑡

𝜏 = 0


(𝜏) 𝑒Γ𝑖𝑚𝑛𝑝𝜏𝑑𝜏 + 𝑎

𝑖1𝑒−Γ𝑖𝑚𝑛𝑝𝑡.

(31)

Application of the initial condition (10) on (26) gives thefollowing:

𝑇𝑖(𝑟, 𝜃, 𝜙, 𝑡 = 0)

= 𝑓𝑖(𝑟, 𝜃, 𝜙)

=

∞

∑𝑚=1

∞

∑𝑛 = 1

∞

∑𝑝= 1

𝑇𝑖𝑚𝑛𝑝 (0) 𝑅𝑖𝑝𝑛 (𝑟) Θ𝑖𝑛 (𝜃)Φ𝑖𝑚 (𝜙) .

(32)

The coefficient 𝑎𝑖1of (31) is found by the use of the orthogo-

nality property for obtained eigenfunctions as follows:

𝑎𝑖1

= 𝑇𝑖𝑚𝑛𝑝 (0)

=∫𝜔

0∫𝜓

0∫𝑟𝑖

𝑟𝑖−1

𝑓𝑖(𝑟, 𝜃, 𝜙) 𝑟2𝑅

𝑖𝑝𝑛(𝑟) Θ𝑖𝑛(𝜃)Φ𝑖𝑚(𝜙) 𝑑𝑟 𝑑𝜃 𝑑𝜙

∫𝜔

0∫𝜓

0∫𝑟𝑖

𝑟𝑖 − 1

𝑟2𝑅2𝑖𝑝𝑛(𝑟) Θ2𝑖𝑛(𝜃)Φ2𝑖𝑚(𝜙) 𝑑𝑟 𝑑𝜃 𝑑𝜙

.

(33)

The solution of differential equation (1) having (2) to (9) asboundary conditions and (10) as initial condition is (26) with(31) as the coefficients.

4. Case Study Problem

We consider a three-layer quarter-sphere (0 ≤ 𝑟 ≤ 𝑟3, 0 ≤

𝜃 ≤ 𝜋/2, and 0 ≤ 𝜙 ≤ 𝜋) which is initially (𝑡 = 0) at uniformunit temperature. For time 𝑡 > 0, thermal convection occurs,from the outer radial surface at 𝑟 = 𝑟

3at (zero) ambient

temperature. However, the 𝜃 = 0, 𝜃 = 𝜋/2, 𝜙 = 0, and 𝜙 = 𝜋surfaces are at uniformand constant zero temperatures.Theseboundary conditions lead to the following: 𝐴 in = 1, 𝐴out =𝑘3, 𝐵in = 0, 𝐵out = ℎ, 𝐶in = 0, and 𝐶out = 0. Additionally, the

uniformly distributed heat source 𝑔𝑖, 𝑖 = 1, . . . , 3, is turned

on in each layer at 𝑡 = 0. The governing differential equationfor the 3D transient heat conduction with heat sources in thisthree-layer quarter-spherical region is as follows:

𝜕2𝑇𝑖

𝜕𝑟2+2

𝑟

𝜕𝑇𝑖

𝜕𝑟+1

𝑟2𝜕2𝑇𝑖

𝜕𝜃2+cot 𝜃𝑟2

𝜕𝑇𝑖

𝜕𝜃+

1

𝑟2sin2𝜃𝜕2𝑇𝑖

𝜕𝜙2+𝑔𝑖

𝑘𝑖

=1

𝛼𝑖

𝜕𝑇𝑖

𝜕𝑡,

𝑇𝑖= 𝑇𝑖(𝑟, 𝜃, 𝜙, 𝑡) , 0 ≤ 𝑟 ≤ 𝑟

3,

𝑟𝑖−1≤ 𝑟 ≤ 𝑟

𝑖, 1 ≤ 𝑖 ≤ 3,

0 ≤ 𝜃 ≤𝜋

2,

0 ≤ 𝜙 ≤ 𝜋,

𝑡 ≥ 0.

(34)

The boundary conditions have the following forms:

𝜕𝑇1

𝜕𝑟(0, 𝜃, 𝜙, 𝑡) = 0, (35)

𝑘3

𝜕𝑇3

𝜕𝑟(𝑟3, 𝜃, 𝜙, 𝑡) + ℎ𝑇

3(𝑟3, 𝜃, 𝜙, 𝑡) = 0, (36)

𝑇𝑖(𝑟,

𝜋

2, 𝜙, 𝑡) = 0, (37)

𝑇𝑖(𝑟, 𝜃, 0, 𝑡) = 0, (38)

𝑇𝑖(𝑟, 𝜃, 𝜋, 𝑡) = 0. (39)

(i) Inner interface of the 𝑖th layer (𝑖 = 2, 3):

𝑇𝑖(𝑟𝑖 − 1, 𝜃, 𝜙, 𝑡) = 𝑇

𝑖 − 1(𝑟𝑖 − 1, 𝜃, 𝜙, 𝑡) ,

𝑘𝑖

𝜕𝑇𝑖

𝜕𝑟(𝑟𝑖 − 1, 𝜃, 𝜙, 𝑡) = 𝑘

𝑖−1

𝜕𝑇𝑖 − 1

𝜕𝑟(𝑟𝑖 − 1, 𝜃, 𝜙, 𝑡) .

(40)

(ii) Outer interface of the 𝑖th layer (𝑖 = 1, 2):

𝑇𝑖(𝑟𝑖, 𝜃, 𝜙, 𝑡) = 𝑇

𝑖 + 1(𝑟𝑖, 𝜃, 𝜙, 𝑡) , (41)

𝑘𝑖

𝜕𝑇𝑖

𝜕𝑟(𝑟𝑖, 𝜃, 𝜙, 𝑡) = 𝑘

𝑖 + 1

𝜕𝑇𝑖 + 1

𝜕𝑟(𝑟𝑖, 𝜃, 𝜙, 𝑡) . (42)

The initial condition is as follows:

𝑇𝑖(𝑟, 𝜃, 𝜙, 0) = 1, 1 ≤ 𝑖 ≤ 3. (43)

According to (25), by the use of the eigenfunction expansionmethod, the following solutions in 𝑧-, 𝑟-, and 𝜃-directions areobtained from the associated eigenvalue problem:

Φ𝑚(𝜙) = 𝑐

1sin ]𝑚𝜙 + 𝑐2cos ]𝑚𝜙,


]𝑚

𝑖𝑛(cos 𝜃) ,

𝑅𝑖𝑝𝑛 (𝑟) =

1

√𝑟[𝑐3𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟) + 𝑐4𝑌𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟)] .

(44)


The eigenvalues are 𝜁2𝑖𝑚𝑛𝑝

= 𝜆2𝑖𝑝𝑛+]2𝑖𝑚.The heat flux continuity

conditions at the interfaces imply the following:

𝜆𝑖𝑝𝑛= 𝜆1𝑝𝑛√

𝛼1

𝛼𝑖

. (45)

The eigenfunctions𝑅𝑖𝑝𝑛(𝑟),Θ

𝑖𝑛(𝜃), andΦ

𝑚(𝜙) in the 𝑟, 𝜃- and

𝜙-directions are obtained by applying the relevant boundaryconditions in each direction. Application of the boundaryconditions in the 𝜙-direction due to (44) gives the following:

Φ𝑚(𝜙) = 𝑐

1sin ]𝑚𝜙 + 𝑐2cos ]𝑚𝜙

→

{{{{{{{

{{{{{{{

{

Φ𝑚 (0) = 0 → 𝑐2 = 0

→ Φ𝑚 (𝜃) = 𝑐1 sin ]𝑚𝜙

Φ𝑚(𝜋) = 0 → 𝑐

1sin ]𝑚𝜋 = 0

𝑐1̸= 0

→ sin ]𝑚𝜋 = 0 = sin (𝑚𝜋) .

(46)

Then the 𝜙-direction eigenvalues and eigenfunction areobtained as follows:

]𝑚= 𝑚, 𝑚 = 1, 2, . . .

Φ𝑚(𝜙) = sin (𝑚𝜙) .

(47)

Using the 𝜃-direction boundary condition, (37), onΘ in (44)results in the following relation:


𝑚

𝑖𝑛(cos 𝜃)

Θ𝑖𝑛(𝜋/2) = 0

→ 𝑃𝑚

𝑖𝑛(cos(𝜋

2)) = 0

→ 𝑃𝑚

𝑖𝑛(0) = 0,

(48)

where 𝑃𝑚𝑖𝑛(0) = 0 is only satisfied when 𝑛 are odd integers;

that is, 𝑛 = 1, 3, 5, . . .. Thus the 𝜃-direction eigenvalues andthe eigenfunction are as follows:

𝑛 = 1, 3, 5, . . .


𝑚

𝑖𝑛(cos 𝜃) .

(49)

Applying the 𝑟-direction boundary conditions, that is, (35)and (36), gives the following:


=1

√𝑟[𝑐3𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟) + 𝑐4𝑌𝑛(𝑛 + 1) + 0.5


𝑖 = 1, 2, 3

→

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

𝑅(0) = 0 → 𝑅 (0) = finite

𝑌𝑛(𝑛 + 1) + 0.5

(0) =∞

→ 𝑐4= 0

→ 𝑅𝑖𝑝𝑛 (𝑟) = 𝑐3

1

√𝑟𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟)

𝑘𝑅 (𝑟3) + ℎ𝑅 (𝑟

3) = 0

→ 𝑘𝑐3

1

√𝑟3𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟3)

− 𝑘𝑐3

1

2𝑟3√𝑟3

𝐽𝑛(𝑛 + 1) + 0.5


+ ℎ𝑐3

1

√𝑟3𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟3) = 0

𝑐3̸= 0

→1

√𝑟3[𝑘𝐽𝑛(𝑛 + 1) + 0.5


+ (ℎ −𝑘

2𝑟3

) 𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟3)] = 0

→ 𝑘𝐽𝑛(𝑛 + 1) + 0.5


+(ℎ −𝑘

2𝑟3

) 𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟3) = 0.

(50)

Thus the 𝑟-direction eigencondition and eigenfunction arederived as (]

𝑚= 𝑚):

𝑘𝐽

𝑛(𝑛 + 1) + 0.5(𝜆𝑖𝑝𝑛𝑟3) + (ℎ −

𝑘

2𝑟3

) 𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟3) = 0,

𝑅𝑖𝑝𝑛 (𝑟) =

1

√𝑟𝐽𝑛(𝑛 + 1) + 0.5

(𝜆𝑖𝑝𝑛𝑟) .

(51)

According to (28) to (33), the coefficients 𝑔𝑖𝑚𝑛𝑝

(𝑡), 𝑎𝑖1, and

Γ𝑖𝑚𝑛𝑝

are obtained as follows:

𝑔𝑖𝑚𝑛𝑝 (𝑡) = (∫

𝜋

0

∫𝜋

0

∫𝑟𝑖

𝑟𝑖 − 1

𝑔𝑖𝑟2 1

√𝑟𝐽𝛽𝑛+ 0.5

(𝜆𝑖𝑝𝑛𝑟) 𝑃𝑚

𝑖𝑛(𝜃)

× sin (𝑚𝜙) 𝑑𝑟 𝑑𝜃 𝑑𝜙)

× (∫𝜋

0

∫𝜋

0

∫𝑟𝑖

𝑟𝑖−1

𝑟𝐽2

𝛽𝑛+ 0.5

(𝜆𝑖𝑝𝑛𝑟) (𝑃𝑚

𝑖𝑛(𝜃))2

× sin2 (𝑚𝜙) 𝑑𝑟 𝑑𝜃 𝑑𝜙)−1


= (𝑔𝑖(∫𝑟𝑖

𝑟𝑖−1

𝑟3/2𝐽𝛽𝑛+ 0.5

(𝜆𝑖𝑝𝑛𝑟) 𝑑𝑟)(∫

𝜋

0

𝑃𝑚

𝑖𝑛(𝜃) 𝑑𝜃)

×(∫𝜋

0

sin (𝑚𝜙) 𝑑𝜙))

× ((∫𝑟𝑖

𝑟𝑖−1

𝑟𝐽2

𝛽𝑛+ 0.5

(𝜆𝑖𝑝𝑛𝑟) 𝑑𝑟) (∫

𝜋

0

(𝑃𝑚

𝑖𝑛(𝜃))2𝑑𝜃)

× (∫𝜋

0

sin2 (𝑚𝜙) 𝑑𝜙))−1

= (𝑔𝑖[(𝜇𝑖𝑛𝑚𝑟) 𝐽𝑚+1

(𝜇𝑖𝑛𝑚𝑟)]𝑟 = 𝑟𝑖

𝑟 = 𝑟𝑖 − 1

× (−1

𝑚cos (𝑚𝜃))

𝜃 =𝜋

𝜃= 0

(−1

𝑚cos (𝑚𝜙))

𝜙=𝜋

𝜙= 0

)

× ([(𝜇𝑖𝑛𝑚𝑟)2

2(𝐽2

𝑚(𝜇𝑖𝑛𝑚𝑟) + 𝐽

2

𝑚+1(𝜇𝑖𝑛𝑚𝑟))]

𝑟 = 𝑟𝑖

𝑟 = 𝑟𝑖 − 1

×𝜋

2×𝐿

2)

−1

= (𝑔𝑖[(𝜇𝑖𝑛𝑚𝑟) 𝐽𝑚+1

(𝜇𝑖𝑛𝑚𝑟)]𝑟 = 𝑟𝑖

𝑟 = 𝑟𝑖 − 1

×1

𝑚(1 − (−1)

𝑚)

×𝐿

𝑙𝜋(1 − (−1)

𝑙))

× ([(𝜇𝑖𝑛𝑚𝑟)2

2(𝐽2

𝑚(𝜇𝑖𝑛𝑚𝑟) + 𝐽

2

𝑚+1(𝜇𝑖𝑛𝑚𝑟))]

𝑟 = 𝑟𝑖

𝑟 = 𝑟𝑖 − 1

×𝜋𝐿

4)

−1

→ 𝑔𝑖𝑚𝑛𝑝

(𝑡) = 𝑔𝑖𝑚𝑛𝑝

= 𝑎𝑖1𝑔𝑖,

Γ𝑖𝑚𝑛𝑝

= (−𝛼𝑖)

× (𝑅𝑖𝑝𝑛(𝑟)


+1

𝑟



+1

𝑟2

Θ𝑖𝑛(𝜃)

Θ𝑖𝑛(𝜃)

+Φ𝑖𝑚(𝜙)

Φ𝑖𝑚(𝜙))

= −𝛼𝑖(𝐽𝑚(𝜆𝑖𝑝𝑛𝑟)

𝐽𝑚(𝜆𝑖𝑝𝑛𝑟)+1

𝑟

𝐽𝑚(𝜆𝑖𝑝𝑛𝑟)

𝐽𝑚(𝜆𝑖𝑝𝑛𝑟)−𝑚2

𝑟2− 𝑚2) .

(52)

Using the form of a triple-series expansion of the obtainedeigenfunctions, that is, (26), the solution is as follows:

𝑇𝑖(𝑟, 𝜃, 𝜙, 𝑡) =

∞

∑𝑚=1

∞

∑𝑛 = 1

∞

∑𝑝= 1

𝑇𝑖𝑚𝑛𝑝 (𝑡)

1

√𝑟𝐽𝛽𝑛+ 0.5

× (𝜆𝑖𝑝𝑛𝑟) 𝑃𝑚

𝑖𝑛(𝜃) sin (𝑚𝜙) ,

(53)

where the coefficient 𝑇𝑖𝑚𝑛𝑝

(𝑡) is attained as follows:

𝑇𝑖𝑚𝑛𝑝 (𝑡) =

𝛼𝑖

𝑘𝑖

𝑒−Γ𝑖𝑚𝑛𝑝𝑡𝑔𝑖𝑚𝑛𝑝

1

Γ𝑖𝑚𝑛𝑝

(𝑒Γ𝑖𝑚𝑛𝑝𝑡− 1) + 𝑎

𝑖1𝑒−Γ𝑖𝑚𝑛𝑝𝑡

=𝛼𝑖𝑔𝑖𝑚𝑛𝑝

𝑘𝑖Γ𝑖𝑚𝑛𝑝

+ (1

𝑔𝑖

−𝛼𝑖

𝑘𝑖Γ𝑖𝑚𝑛𝑝

)𝑔𝑖𝑚𝑛𝑝

𝑒−Γ𝑖𝑚𝑛𝑝𝑡.

(54)

Therefore, the solution of (34) having (35) to (42) as boundaryconditions and (43) as initial condition is (53) with (54) as thecoefficients.

5. Conclusions

The exact analytical solution, that is, transient temperaturedistribution, is derived for the 3D transient heat conductionproblem in amultilayered sphere using eigenfunction expan-sion method. Time-dependent and nonuniform volumetricheat generation is considered in each radial layer. Third kindnonhomogeneous boundary conditions are applied in theradial direction but the first or second kind homogenousboundary conditions are used in the angular and azimuthaldirections. The heat conduction in a three-layer quarter-sphere is solved as a case study problem and the temperaturedistribution is found.

Nomenclature

𝐴 in, 𝐵in, 𝐶in: Coefficients in (2)𝐴out, 𝐵out, 𝐶out: Coefficients in (3)𝑓𝑖(𝑟, 𝜃, 𝜙): Initial temperature distribution in the

𝑖th layer at 𝑡 = 0𝑔𝑖(𝑟, 𝜃, 𝜙, 𝑡): Volumetric heat source distribution in

the 𝑖th layer𝑔𝑖𝑚𝑛𝑝

: Coefficient in series expansion for heatsource (Equation (27))

ℎ: Outer surface heat transfer coefficient𝐽]𝑚: Bessel function of the first kind of order

]𝑚

𝑘𝑖: Thermal conductivity of the 𝑖th layer

𝑟: Radial coordinate𝑟𝑖: Outer radius for the 𝑖th layer𝑅𝑖𝑝𝑛(𝑟): Radial eigenfunctions for the 𝑖th layer

𝑡: Time𝑇𝑖(𝑟, 𝜃, 𝜙, 𝑡): Temperature distribution for the 𝑖th

layer𝑇𝑖𝑚𝑛𝑝

: Coefficient in general solution(Equation (26)) dependent on initialcondition

𝑌]𝑚: Bessel function of the second kind oforder ]

𝑚.

Greek Symbols

𝛼𝑖: Thermal diffusivity of the 𝑖th layer

𝜃: Azimuthal coordinateΦ𝑖𝑚(𝜙): Eigenfunctions in the 𝜙-direction


Θ𝑖𝑛(𝜃): Eigenfunctions in the angular direction

𝜆𝑖𝑝𝑛: Radial eigenvalues

]𝑚: Eigenvalues in the 𝜙-direction

𝜔: Angle subtended by the multilayers inthe 𝜙-direction

𝜓: Angle subtended by the multilayers inthe 𝜃-direction

Γ𝑖𝑚𝑛𝑝

: Coefficient in (30).

Subscripts and Superscripts

𝑖: Layer or interface number: Differentiation.

Conflict of Interests

The author of the paper declares that there is no conflict ofinterests regarding the publication of this paper.

References

[1] H. Salt, “Transient heat conduction in a two-dimensional com-posite slab. I.Theoretical development of temperatures modes,”International Journal of Heat and Mass Transfer, vol. 26, no. 11,pp. 1611–1616, 1983.

[2] M. D. Mikhailov and M. N. Özişik, “Transient conduction ina three-dimensional composite slab,” International Journal ofHeat and Mass Transfer, vol. 29, no. 2, pp. 340–342, 1986.

[3] A. Haji-Sheikh and J. V. Beck, “Temperature solution in multi-dimensional multi-layer bodies,” International Journal of Heatand Mass Transfer, vol. 45, no. 9, pp. 1865–1877, 2002.

[4] F. de Monte, “Transverse eigenproblem of steady-state heatconduction for multi- dimensional two-layered slabs withautomatic computation of eigenvalues,” International Journal ofHeat and Mass Transfer, vol. 47, no. 2, pp. 191–201, 2004.

[5] F. de Monte, “Multi-layer transient heat conduction using tran-sition time scales,” International Journal ofThermal Sciences, vol.45, no. 9, pp. 882–892, 2006.

[6] X. Lu, P. Tervola, andM.Viljanen, “Transient analytical solutionto heat conduction in composite circular cylinder,” InternationalJournal of Heat and Mass Transfer, vol. 49, no. 1-2, pp. 341–348,2006.

[7] X. Lu and M. Viljanen, “An analytical method to solve heatconduction in layered spheres with time-dependent boundaryconditions,” Physics Letters Section A: General, Atomic and SolidState Physics, vol. 351, no. 4-5, pp. 274–282, 2006.

[8] S. Singh, P. K. Jain, and Rizwan-uddin, “Analytical solution totransient heat conduction in polar coordinates with multiplelayers in radial direction,” International Journal of ThermalSciences, vol. 47, no. 3, pp. 261–273, 2008.

[9] S. Singh and P. K. Jain, “Finite integral transform method tosolve asymmetric heat conduction in a multilayer annulus withtime-dependent boundary conditions,” Nuclear Engineeringand Design, vol. 241, no. 1, pp. 144–154, 2011.

[10] P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution totransient asymmetric heat conduction in a multilayer annulus,”Journal of Heat Transfer, vol. 131, no. 1, pp. 1–7, 2009.

[11] P. K. Jain and S. Singh, “An exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical

coordinates,” International Journal of Heat and Mass Transfer,vol. 53, no. 9-10, pp. 2133–2142, 2010.

[12] N. Dalir and S. S. Nourazar, “Analytical solution of theproblem on the three-dimensional transient heat conductionin a multilayer cylinder,” Journal of Engineering Physics andThermophysics, vol. 87, no. 1, pp. 89–97, 2014.

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