research article exact analytical solution for 3d time ...solutions for d multilayer transient heat...
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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 ยฉ 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 includeGreenโs 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 Oฬzisฬงik [2], usingthe orthogonal expansion approach, solved the problem oftransient three-dimensional heat conduction in a compositeCartesian medium. Haji-Sheikh and Beck [3] used Greenโsfunction 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
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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 ๐ = constant and ๐ = constant. However,nonhomogeneous boundary conditions of the third kind(convection) [11] are used in the ๐-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 (๐-๐-๐coordinates) or part-sphericalmultilayer geometries. Second,volumetric internal heat sources of nonuniform and time-dependent (๐, ๐, ๐ and ๐ก-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 ๐ and ๐-directions, only the firstor second kind of homogeneous boundary condition can beapplied.
2. Mathematical Formulation
A ๐-layer composite spherical slab (๐0โค ๐ โค ๐
๐, 0 โค ๐ โค
๐, and 0 โค ๐ โค ๐) is considered. All the layers have perfectthermal contact and are presumed to be isotropic in thermalproperties. ๐ผ
๐and ๐
๐are the temperature independent ther-
mal diffusivity and thermal conductivity of the ๐th layer. At๐ก = 0, the ๐th layer is at a specified temperature ๐
๐(๐, ๐, ๐)
and time dependent heat sources ๐๐(๐, ๐, ๐, ๐ก) are switched
on in each radial layer. For ๐ก > 0, homogenous boundaryconditions of first or second kind are applied to the angularsurfaces of ๐ = 0 and ๐ = ๐ and azimuthal surfaces of ๐ = 0and ๐ = ๐. For the inner (๐ = 1, ๐ = ๐
0) and the outer
(๐ = ๐, ๐ = ๐๐) 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:
๐2๐๐
๐๐2+2
๐
๐๐๐
๐๐+1
๐2๐2๐๐
๐๐2+cot ๐๐2
๐๐๐
๐๐+
1
๐2sin2๐๐2๐๐
๐๐2
+๐๐(๐, ๐, ๐, ๐ก)
๐๐
=1
๐ผ๐
๐๐๐
๐๐ก,
๐๐= ๐๐(๐, ๐, ๐, ๐ก) , ๐
0โค ๐ โค ๐
๐,
๐๐โ1โค ๐ โค ๐
๐, 1 โค ๐ โค ๐,
0 โค ๐ โค ๐, ๐ < ๐,
0 โค ๐ โค ๐, ๐ < 2๐,
๐ก โฅ 0.
(1)
The boundary conditions are as follows.
(i) Inner surface of 1st layer (๐ = 1):
๐ด in๐๐1
๐๐(๐0, ๐, ๐, ๐ก) + ๐ตin๐1 (๐0, ๐, ๐, ๐ก) = ๐ถin. (2)
(ii) Outer surface of ๐th layer (๐ = ๐):
๐ดout๐๐๐
๐๐(๐๐, ๐, ๐, ๐ก) + ๐ตout๐๐ (๐๐, ๐, ๐, ๐ก) = ๐ถout. (3)
(iii) ๐ = ๐ surface (๐ = 1, 2, . . . , ๐):
๐๐(๐, ๐ = ๐, ๐, ๐ก) = 0 or
๐๐๐
๐๐(๐, ๐ = ๐, ๐, ๐ก) = 0,
๐ = 1, . . . , ๐.
(4)
(iv) ๐ = 0 surface (๐ = 1, 2, . . . , ๐):
๐๐(๐, ๐, ๐ = 0, ๐ก) = 0 or
๐๐๐
๐๐(๐, ๐, ๐ = 0, ๐ก) = 0,
๐ = 1, . . . , ๐.
(5)
(v) ๐ = ๐ surface (๐ = 1, 2, . . . , ๐):
๐๐(๐, ๐, ๐ = ๐, ๐ก) = 0 or
๐๐๐
๐๐(๐, ๐, ๐ = ๐, ๐ก) = 0,
๐ = 1, . . . , ๐.
(6)
(vi) Inner interface of the ๐th layer (๐ = 2, . . . , ๐):
๐๐(๐๐โ1, ๐, ๐, ๐ก) = ๐
๐โ1(๐๐โ1, ๐, ๐, ๐ก) ๐ = 2, . . . , ๐,
๐๐
๐๐๐
๐๐(๐๐โ1, ๐, ๐, ๐ก) = ๐
๐โ1
๐๐๐โ1
๐๐(๐๐โ1, ๐, ๐, ๐ก) ๐ = 2, . . . , ๐.
(7)
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International Scholarly Research Notices 3
(vii) Outer interface of the ๐th layer (๐ = 1, . . . , ๐ โ 1):
๐๐(๐๐, ๐, ๐, ๐ก) = ๐
๐+1(๐๐, ๐, ๐, ๐ก) ๐ = 1, . . . , ๐ โ 1, (8)
๐๐
๐๐๐
๐๐(๐๐, ๐, ๐, ๐ก) = ๐
๐+1
๐๐๐+1
๐๐(๐๐, ๐, ๐, ๐ก) ๐ = 1, . . . , ๐ โ 1.
(9)
The initial condition is as follows:
๐๐(๐, ๐, ๐, ๐ก = 0) = ๐
๐(๐, ๐, ๐) , 1 โค ๐ โค ๐. (10)
It is worth mentioning that, at ๐ = ๐0and ๐ = ๐
๐, 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 (๐
0= 0) for
multilayered sphere ismodeled by assigning zero values to๐ตinand ๐ถin 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 (โ2๐ = โ๐2๐),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:
โ2๐๐= โ๐2๐๐โ
๐2๐๐
๐๐2+2
๐
๐๐๐
๐๐+1
๐2๐2๐๐
๐๐2
+cot ๐๐2
๐๐๐
๐๐+
1
๐2sin2๐๐2๐๐
๐๐2
= โ๐2๐๐.
(11)
Using the method of separation of variables, (11) is solved asfollows:
๐๐(๐, ๐, ๐) = ๐
๐(๐) ฮ๐(๐)ฮฆ๐(๐) , (12)
๐ ๐ฮ๐ฮฆ๐
๐ ๐ฮ๐ฮฆ๐
+2
๐
๐ ๐ฮ๐ฮฆ๐
๐ ๐ฮ๐ฮฆ๐
+1
๐2
๐ ๐ฮ๐ฮฆ๐
๐ ๐ฮ๐ฮฆ๐
+cot ๐๐2
๐ ๐ฮ๐ฮฆ๐
๐ ๐ฮ๐ฮฆ๐
+1
๐2sin2๐๐ ๐ฮ๐ฮฆ๐
๐ ๐ฮ๐ฮฆ๐
= โ๐2๐ ๐ฮ๐ฮฆ๐
๐ ๐ฮ๐ฮฆ๐
,
(13)
๐ ๐
๐ ๐
+2
๐
๐ ๐
๐ ๐
+1
๐2
ฮ๐
ฮ๐
+cot ๐๐2
ฮ๐
ฮ๐
+1
๐2sin2๐ฮฆ๐
ฮฆ๐
= โ๐2, (14)
๐ ๐
๐ ๐
+2
๐
๐ ๐
๐ ๐
+1
๐2
ฮ๐
ฮ๐
+cot ๐๐2
ฮ๐
ฮ๐
= โ1
๐2sin2๐ฮฆ๐
ฮฆ๐
โ ๐2, (15)
sin2๐(๐2๐ ๐
๐ ๐
+ 2๐๐ ๐
๐ ๐
+ฮ๐
ฮ๐
+ cot ๐ฮ๐
ฮ๐
+ ๐2๐2)
= โฮฆ๐
ฮฆ๐
โ = +]2,
(16)
ฮฆ
๐+ ]2๐๐ฮฆ๐= 0 โ ฮฆ
๐๐(๐) = ๐
1sin ]๐๐๐ + ๐2cos ]๐๐๐,
๐2๐ ๐
๐ ๐
+ 2๐๐ ๐
๐ ๐
+ฮ๐
ฮ๐
+ cot ๐ฮ๐
ฮ๐
+ ๐2๐2=
]2
sin2๐,
(17)
โ ๐2๐ ๐
๐ ๐
+ 2๐๐ ๐
๐ ๐
+ ๐2๐2= โ
ฮ๐
ฮ๐
โ cot ๐ฮ๐
ฮ๐
+]2
sin2๐
= +๐ฝ2,
(18)
๐2๐
๐+ 2๐๐
๐+ (๐2
๐๐๐๐2โ ๐ฝ2
๐) ๐ ๐= 0
โ ๐ ๐๐๐(๐) =
1
โ๐[๐3๐ฝ๐ฝ๐+0.5
(๐๐๐๐๐)
+ ๐4๐๐ฝ๐+0.5
(๐๐๐๐๐)] ,
ฮ
๐+ (cot ๐)ฮ
๐+ (๐ฝ2โ
]2
sin2๐)ฮ๐= 0.
(19)
By using the following change of variable:
๐ = cos ๐ โ 1 โ ๐2 = sin2๐ (20)the first and second derivatives of ฮ with respect to ๐ areobtained as follows:
ฮ=๐ฮ๐
๐๐=๐ฮ๐
๐๐
๐๐
๐๐= โ sin ๐
๐ฮ๐
๐๐,
ฮ
๐=๐2ฮ๐
๐๐2=๐
๐๐(๐ฮ๐
๐๐) =
๐
๐๐(โ sin ๐
๐ฮ๐
๐๐)
= โ cos ๐๐ฮ๐
๐๐โ sin ๐ ๐
๐๐(๐ฮ๐
๐๐)
= sin2๐๐2ฮ๐
๐๐2โ cos ๐
๐ฮ๐
๐๐.
(21)
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4 International Scholarly Research Notices
Substituting (21) in (19) results in the following:
ฮ
๐+ (cot ๐)ฮ
๐+ (๐ฝ2โ
]2
sin2๐)ฮ๐= 0
โ sin2๐๐2ฮ๐
๐๐2โ cos ๐
๐ฮ๐
๐๐
+ (cos ๐sin ๐
)(โ sin ๐๐ฮ๐
๐๐) + (๐ฝ
2โ
]2
sin2๐)ฮ๐= 0
โ (1 โ ๐2)๐2ฮ๐
๐๐2โ 2๐
๐ฮ๐
๐๐+ (๐ฝ2
๐โ
]2๐๐
1 โ ๐2)ฮ๐= 0.
(22)
If ๐ฝ2๐= ๐(๐ + 1), (22) is the associated Legendre equation. Its
solution is written as follows:
ฮ๐๐(๐) = ๐
5๐]๐๐
๐๐(๐) + ๐
6๐
]๐๐
๐๐(๐)
โ ฮ๐๐ (๐) = ๐5๐
]๐๐
๐๐(cos ๐) + ๐6๐
]๐๐
๐๐(cos ๐)
๐]๐๐
๐๐(cos 0) =๐]๐๐
๐๐(1) =โโ๐
6= 0
โ ฮ๐๐(๐) = ๐
]๐๐
๐๐(cos ๐) .
(23)
The problem eigenvalues are ๐2๐๐๐๐
= ๐2๐๐๐+ ]2๐๐. It should be
stated that the heat fluxes continuity at the interfaces of theradial layers gives the following:
ฮฆ๐๐= ฮฆ๐, ]
๐๐= ]๐,
๐๐๐๐= ๐1๐๐โ
๐ผ1
๐ผ๐
.
(24)
The eigenfunctions ๐ ๐๐๐(๐), ฮ
๐๐(๐), and ฮฆ
๐๐(๐) in ๐-, ๐-, and
๐-directions are derived by applying the boundary conditionsin each direction in the following equation:
๐ ๐๐๐(๐) =
1
โ๐[๐3๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐) + ๐4๐๐(๐ + 1) + 0.5
(๐๐๐๐๐)] ,
ฮ๐๐(๐) = ๐
]๐
๐๐(cos ๐) ,
ฮฆ๐(๐) = ๐
1sin ]๐๐ + ๐2cos ]๐๐.
(25)
It is assumed that the solution of the problem is in the formof a triple-series expansion of the derived eigenfunctions asfollows:
๐๐(๐, ๐, ๐, ๐ก) =
โ
โ๐=1
โ
โ๐ = 1
โ
โ๐= 1
๐๐๐๐๐ (๐ก) ๐ ๐๐๐ (๐) ฮ๐๐ (๐)ฮฆ๐ (๐) .
(26)The heat source term is also written as a triple-series expan-sion of the eigenfunctions such that
๐๐(๐, ๐, ๐, ๐ก) =
โ
โ๐=1
โ
โ๐ = 1
โ
โ๐= 1
๐๐๐๐๐
(๐ก) ๐ ๐๐๐(๐) ฮ๐๐(๐)ฮฆ๐(๐) ,
(27)
where the coefficient ๐๐๐๐๐
(๐ก) is obtained by the use of theorthogonality property as follows:
๐๐๐๐๐
(๐ก) = (โซ๐
0
โซ๐
0
โซ๐๐
๐๐ โ 1
๐๐(๐, ๐, ๐, ๐ก) ๐
2
ร ๐ ๐๐๐(๐) ฮ๐๐(๐)ฮฆ๐(๐) ๐๐ ๐๐ ๐๐)
ร (โซ๐
0
โซ๐
0
โซ๐๐
๐๐ โ 1
๐2๐ 2
๐๐๐(๐) ฮ2
๐๐(๐)
ร ฮฆ2
๐(๐) ๐๐ ๐๐ ๐๐)
โ1
.
(28)
Substitution of (26) and (27) in (1) results in the following:
๐๐๐๐๐
(๐ก) ๐
๐๐๐(๐) ฮ๐๐(๐)ฮฆ๐๐(๐) +
2
๐๐๐๐๐๐
(๐ก) ๐
๐๐๐(๐) ฮ๐๐(๐)ฮฆ๐๐(๐) +
1
๐2๐๐๐๐๐
(๐ก) ๐ ๐๐๐(๐) ฮ
๐๐(๐)ฮฆ๐๐(๐)
+cot ๐๐2
๐๐๐๐๐
(๐ก) ๐ ๐๐๐(๐) ฮ
๐๐(๐)ฮฆ๐๐(๐) +
1
๐2sin2๐๐๐๐๐๐
(๐ก) ๐ ๐๐๐(๐) ฮ๐๐(๐)ฮฆ
๐๐(๐)
+1
๐๐
๐๐๐๐๐
(๐ก) ๐ ๐๐๐(๐) ฮ๐๐(๐)ฮฆ๐๐(๐) =
1
๐ผ๐
๐
๐๐๐๐(๐ก) ๐ ๐๐๐(๐) ฮ๐๐(๐)ฮฆ๐๐(๐) ,
(29)
โ๐๐๐๐๐๐ (๐ก)
๐๐ก+ (โ๐ผ
๐)(
๐ ๐๐๐(๐)
๐ ๐๐๐ (๐)
+2
๐
๐ ๐๐๐(๐)
๐ ๐๐๐ (๐)
+1
๐2
ฮ๐๐(๐)
ฮ๐๐ (๐)
+cot ๐๐2
ฮ๐๐(๐)
ฮ๐๐ (๐)
+1
๐2sin2๐ฮฆ๐๐(๐)
ฮฆ๐๐(๐)
)โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
= ฮ๐๐๐๐
๐๐๐๐๐
(๐ก) =๐ผ๐
๐๐
๐๐๐๐๐
(๐ก) . (30)
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International Scholarly Research Notices 5
Equation (30) is a first-order nonhomogeneous ODE and hasthe following solution:
๐๐๐๐๐
(๐ก) =๐ผ๐
๐๐
๐โฮ๐๐๐๐๐กโซ๐ = ๐ก
๐ = 0
๐๐๐๐๐
(๐) ๐ฮ๐๐๐๐๐๐๐ + ๐
๐1๐โฮ๐๐๐๐๐ก.
(31)
Application of the initial condition (10) on (26) gives thefollowing:
๐๐(๐, ๐, ๐, ๐ก = 0)
= ๐๐(๐, ๐, ๐)
=
โ
โ๐=1
โ
โ๐ = 1
โ
โ๐= 1
๐๐๐๐๐ (0) ๐ ๐๐๐ (๐) ฮ๐๐ (๐)ฮฆ๐๐ (๐) .
(32)
The coefficient ๐๐1of (31) is found by the use of the orthogo-
nality property for obtained eigenfunctions as follows:
๐๐1
= ๐๐๐๐๐ (0)
=โซ๐
0โซ๐
0โซ๐๐
๐๐โ1
๐๐(๐, ๐, ๐) ๐2๐
๐๐๐(๐) ฮ๐๐(๐)ฮฆ๐๐(๐) ๐๐ ๐๐ ๐๐
โซ๐
0โซ๐
0โซ๐๐
๐๐ โ 1
๐2๐ 2๐๐๐(๐) ฮ2๐๐(๐)ฮฆ2๐๐(๐) ๐๐ ๐๐ ๐๐
.
(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 โค ๐ โค ๐3, 0 โค
๐ โค ๐/2, and 0 โค ๐ โค ๐) which is initially (๐ก = 0) at uniformunit temperature. For time ๐ก > 0, thermal convection occurs,from the outer radial surface at ๐ = ๐
3at (zero) ambient
temperature. However, the ๐ = 0, ๐ = ๐/2, ๐ = 0, and ๐ = ๐surfaces are at uniformand constant zero temperatures.Theseboundary conditions lead to the following: ๐ด in = 1, ๐ดout =๐3, ๐ตin = 0, ๐ตout = โ, ๐ถin = 0, and ๐ถout = 0. Additionally, the
uniformly distributed heat source ๐๐, ๐ = 1, . . . , 3, is turned
on in each layer at ๐ก = 0. The governing differential equationfor the 3D transient heat conduction with heat sources in thisthree-layer quarter-spherical region is as follows:
๐2๐๐
๐๐2+2
๐
๐๐๐
๐๐+1
๐2๐2๐๐
๐๐2+cot ๐๐2
๐๐๐
๐๐+
1
๐2sin2๐๐2๐๐
๐๐2+๐๐
๐๐
=1
๐ผ๐
๐๐๐
๐๐ก,
๐๐= ๐๐(๐, ๐, ๐, ๐ก) , 0 โค ๐ โค ๐
3,
๐๐โ1โค ๐ โค ๐
๐, 1 โค ๐ โค 3,
0 โค ๐ โค๐
2,
0 โค ๐ โค ๐,
๐ก โฅ 0.
(34)
The boundary conditions have the following forms:
๐๐1
๐๐(0, ๐, ๐, ๐ก) = 0, (35)
๐3
๐๐3
๐๐(๐3, ๐, ๐, ๐ก) + โ๐
3(๐3, ๐, ๐, ๐ก) = 0, (36)
๐๐(๐,
๐
2, ๐, ๐ก) = 0, (37)
๐๐(๐, ๐, 0, ๐ก) = 0, (38)
๐๐(๐, ๐, ๐, ๐ก) = 0. (39)
(i) Inner interface of the ๐th layer (๐ = 2, 3):
๐๐(๐๐ โ 1, ๐, ๐, ๐ก) = ๐
๐ โ 1(๐๐ โ 1, ๐, ๐, ๐ก) ,
๐๐
๐๐๐
๐๐(๐๐ โ 1, ๐, ๐, ๐ก) = ๐
๐โ1
๐๐๐ โ 1
๐๐(๐๐ โ 1, ๐, ๐, ๐ก) .
(40)
(ii) Outer interface of the ๐th layer (๐ = 1, 2):
๐๐(๐๐, ๐, ๐, ๐ก) = ๐
๐ + 1(๐๐, ๐, ๐, ๐ก) , (41)
๐๐
๐๐๐
๐๐(๐๐, ๐, ๐, ๐ก) = ๐
๐ + 1
๐๐๐ + 1
๐๐(๐๐, ๐, ๐, ๐ก) . (42)
The initial condition is as follows:
๐๐(๐, ๐, ๐, 0) = 1, 1 โค ๐ โค 3. (43)
According to (25), by the use of the eigenfunction expansionmethod, the following solutions in ๐ง-, ๐-, and ๐-directions areobtained from the associated eigenvalue problem:
ฮฆ๐(๐) = ๐
1sin ]๐๐ + ๐2cos ]๐๐,
ฮ๐๐(๐) = ๐
]๐
๐๐(cos ๐) ,
๐ ๐๐๐ (๐) =
1
โ๐[๐3๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐) + ๐4๐๐(๐ + 1) + 0.5
(๐๐๐๐๐)] .
(44)
-
6 International Scholarly Research Notices
The eigenvalues are ๐2๐๐๐๐
= ๐2๐๐๐+]2๐๐.The heat flux continuity
conditions at the interfaces imply the following:
๐๐๐๐= ๐1๐๐โ
๐ผ1
๐ผ๐
. (45)
The eigenfunctions๐ ๐๐๐(๐),ฮ
๐๐(๐), andฮฆ
๐(๐) in the ๐, ๐- and
๐-directions are obtained by applying the relevant boundaryconditions in each direction. Application of the boundaryconditions in the ๐-direction due to (44) gives the following:
ฮฆ๐(๐) = ๐
1sin ]๐๐ + ๐2cos ]๐๐
โ
{{{{{{{
{{{{{{{
{
ฮฆ๐ (0) = 0 โ ๐2 = 0
โ ฮฆ๐ (๐) = ๐1 sin ]๐๐
ฮฆ๐(๐) = 0 โ ๐
1sin ]๐๐ = 0
๐1ฬธ= 0
โ sin ]๐๐ = 0 = sin (๐๐) .
(46)
Then the ๐-direction eigenvalues and eigenfunction areobtained as follows:
]๐= ๐, ๐ = 1, 2, . . .
ฮฆ๐(๐) = sin (๐๐) .
(47)
Using the ๐-direction boundary condition, (37), onฮ in (44)results in the following relation:
ฮ๐๐(๐) = ๐
๐
๐๐(cos ๐)
ฮ๐๐(๐/2) = 0
โ ๐๐
๐๐(cos(๐
2)) = 0
โ ๐๐
๐๐(0) = 0,
(48)
where ๐๐๐๐(0) = 0 is only satisfied when ๐ are odd integers;
that is, ๐ = 1, 3, 5, . . .. Thus the ๐-direction eigenvalues andthe eigenfunction are as follows:
๐ = 1, 3, 5, . . .
ฮ๐๐(๐) = ๐
๐
๐๐(cos ๐) .
(49)
Applying the ๐-direction boundary conditions, that is, (35)and (36), gives the following:
๐ ๐๐๐ (๐)
=1
โ๐[๐3๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐) + ๐4๐๐(๐ + 1) + 0.5
(๐๐๐๐๐)] ,
๐ = 1, 2, 3
โ
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{
๐ (0) = 0 โ ๐ (0) = finite
๐๐(๐ + 1) + 0.5
(0) =โ
โ ๐4= 0
โ ๐ ๐๐๐ (๐) = ๐3
1
โ๐๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐)
๐๐ (๐3) + โ๐ (๐
3) = 0
โ ๐๐3
1
โ๐3๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐3)
โ ๐๐3
1
2๐3โ๐3
๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐3)
+ โ๐3
1
โ๐3๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐3) = 0
๐3ฬธ= 0
โ1
โ๐3[๐๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐3)
+ (โ โ๐
2๐3
) ๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐3)] = 0
โ ๐๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐3)
+(โ โ๐
2๐3
) ๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐3) = 0.
(50)
Thus the ๐-direction eigencondition and eigenfunction arederived as (]
๐= ๐):
๐๐ฝ
๐(๐ + 1) + 0.5(๐๐๐๐๐3) + (โ โ
๐
2๐3
) ๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐3) = 0,
๐ ๐๐๐ (๐) =
1
โ๐๐ฝ๐(๐ + 1) + 0.5
(๐๐๐๐๐) .
(51)
According to (28) to (33), the coefficients ๐๐๐๐๐
(๐ก), ๐๐1, and
ฮ๐๐๐๐
are obtained as follows:
๐๐๐๐๐ (๐ก) = (โซ
๐
0
โซ๐
0
โซ๐๐
๐๐ โ 1
๐๐๐2 1
โ๐๐ฝ๐ฝ๐+ 0.5
(๐๐๐๐๐) ๐๐
๐๐(๐)
ร sin (๐๐) ๐๐ ๐๐ ๐๐)
ร (โซ๐
0
โซ๐
0
โซ๐๐
๐๐โ1
๐๐ฝ2
๐ฝ๐+ 0.5
(๐๐๐๐๐) (๐๐
๐๐(๐))2
ร sin2 (๐๐) ๐๐ ๐๐ ๐๐)โ1
-
International Scholarly Research Notices 7
= (๐๐(โซ๐๐
๐๐โ1
๐3/2๐ฝ๐ฝ๐+ 0.5
(๐๐๐๐๐) ๐๐)(โซ
๐
0
๐๐
๐๐(๐) ๐๐)
ร(โซ๐
0
sin (๐๐) ๐๐))
ร ((โซ๐๐
๐๐โ1
๐๐ฝ2
๐ฝ๐+ 0.5
(๐๐๐๐๐) ๐๐) (โซ
๐
0
(๐๐
๐๐(๐))2๐๐)
ร (โซ๐
0
sin2 (๐๐) ๐๐))โ1
= (๐๐[(๐๐๐๐๐) ๐ฝ๐+1
(๐๐๐๐๐)]๐ = ๐๐
๐ = ๐๐ โ 1
ร (โ1
๐cos (๐๐))
๐ =๐
๐= 0
(โ1
๐cos (๐๐))
๐=๐
๐= 0
)
ร ([(๐๐๐๐๐)2
2(๐ฝ2
๐(๐๐๐๐๐) + ๐ฝ
2
๐+1(๐๐๐๐๐))]
๐ = ๐๐
๐ = ๐๐ โ 1
ร๐
2ร๐ฟ
2)
โ1
= (๐๐[(๐๐๐๐๐) ๐ฝ๐+1
(๐๐๐๐๐)]๐ = ๐๐
๐ = ๐๐ โ 1
ร1
๐(1 โ (โ1)
๐)
ร๐ฟ
๐๐(1 โ (โ1)
๐))
ร ([(๐๐๐๐๐)2
2(๐ฝ2
๐(๐๐๐๐๐) + ๐ฝ
2
๐+1(๐๐๐๐๐))]
๐ = ๐๐
๐ = ๐๐ โ 1
ร๐๐ฟ
4)
โ1
โ ๐๐๐๐๐
(๐ก) = ๐๐๐๐๐
= ๐๐1๐๐,
ฮ๐๐๐๐
= (โ๐ผ๐)
ร (๐ ๐๐๐(๐)
๐ ๐๐๐(๐)
+1
๐
๐ ๐๐๐(๐)
๐ ๐๐๐(๐)
+1
๐2
ฮ๐๐(๐)
ฮ๐๐(๐)
+ฮฆ๐๐(๐)
ฮฆ๐๐(๐))
= โ๐ผ๐(๐ฝ๐(๐๐๐๐๐)
๐ฝ๐(๐๐๐๐๐)+1
๐
๐ฝ๐(๐๐๐๐๐)
๐ฝ๐(๐๐๐๐๐)โ๐2
๐2โ ๐2) .
(52)
Using the form of a triple-series expansion of the obtainedeigenfunctions, that is, (26), the solution is as follows:
๐๐(๐, ๐, ๐, ๐ก) =
โ
โ๐=1
โ
โ๐ = 1
โ
โ๐= 1
๐๐๐๐๐ (๐ก)
1
โ๐๐ฝ๐ฝ๐+ 0.5
ร (๐๐๐๐๐) ๐๐
๐๐(๐) sin (๐๐) ,
(53)
where the coefficient ๐๐๐๐๐
(๐ก) is attained as follows:
๐๐๐๐๐ (๐ก) =
๐ผ๐
๐๐
๐โฮ๐๐๐๐๐ก๐๐๐๐๐
1
ฮ๐๐๐๐
(๐ฮ๐๐๐๐๐กโ 1) + ๐
๐1๐โฮ๐๐๐๐๐ก
=๐ผ๐๐๐๐๐๐
๐๐ฮ๐๐๐๐
+ (1
๐๐
โ๐ผ๐
๐๐ฮ๐๐๐๐
)๐๐๐๐๐
๐โฮ๐๐๐๐๐ก.
(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
๐ด in, ๐ตin, ๐ถin: Coefficients in (2)๐ดout, ๐ตout, ๐ถout: Coefficients in (3)๐๐(๐, ๐, ๐): Initial temperature distribution in the
๐th layer at ๐ก = 0๐๐(๐, ๐, ๐, ๐ก): Volumetric heat source distribution in
the ๐th layer๐๐๐๐๐
: Coefficient in series expansion for heatsource (Equation (27))
โ: Outer surface heat transfer coefficient๐ฝ]๐: Bessel function of the first kind of order
]๐
๐๐: Thermal conductivity of the ๐th layer
๐: Radial coordinate๐๐: Outer radius for the ๐th layer๐ ๐๐๐(๐): Radial eigenfunctions for the ๐th layer
๐ก: Time๐๐(๐, ๐, ๐, ๐ก): Temperature distribution for the ๐th
layer๐๐๐๐๐
: Coefficient in general solution(Equation (26)) dependent on initialcondition
๐]๐: Bessel function of the second kind oforder ]
๐.
Greek Symbols
๐ผ๐: Thermal diffusivity of the ๐th layer
๐: Azimuthal coordinateฮฆ๐๐(๐): Eigenfunctions in the ๐-direction
-
8 International Scholarly Research Notices
ฮ๐๐(๐): Eigenfunctions in the angular direction
๐๐๐๐: Radial eigenvalues
]๐: Eigenvalues in the ๐-direction
๐: Angle subtended by the multilayers inthe ๐-direction
๐: Angle subtended by the multilayers inthe ๐-direction
ฮ๐๐๐๐
: Coefficient in (30).
Subscripts and Superscripts
๐: 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, โTransient heat conduction in a two-dimensional com-posite slab. I.Theoretical development of temperatures modes,โInternational Journal of Heat and Mass Transfer, vol. 26, no. 11,pp. 1611โ1616, 1983.
[2] M. D. Mikhailov and M. N. Oฬzisฬงik, โTransient conduction ina three-dimensional composite slab,โ International Journal ofHeat and Mass Transfer, vol. 29, no. 2, pp. 340โ342, 1986.
[3] A. Haji-Sheikh and J. V. Beck, โTemperature solution in multi-dimensional multi-layer bodies,โ International Journal of Heatand Mass Transfer, vol. 45, no. 9, pp. 1865โ1877, 2002.
[4] F. de Monte, โTransverse eigenproblem of steady-state heatconduction for multi- dimensional two-layered slabs withautomatic computation of eigenvalues,โ International Journal ofHeat and Mass Transfer, vol. 47, no. 2, pp. 191โ201, 2004.
[5] F. de Monte, โMulti-layer transient heat conduction using tran-sition time scales,โ International Journal ofThermal Sciences, vol.45, no. 9, pp. 882โ892, 2006.
[6] X. Lu, P. Tervola, andM.Viljanen, โTransient analytical solutionto heat conduction in composite circular cylinder,โ InternationalJournal of Heat and Mass Transfer, vol. 49, no. 1-2, pp. 341โ348,2006.
[7] X. Lu and M. Viljanen, โAn analytical method to solve heatconduction in layered spheres with time-dependent boundaryconditions,โ Physics Letters Section A: General, Atomic and SolidState Physics, vol. 351, no. 4-5, pp. 274โ282, 2006.
[8] S. Singh, P. K. Jain, and Rizwan-uddin, โAnalytical solution totransient heat conduction in polar coordinates with multiplelayers in radial direction,โ International Journal of ThermalSciences, vol. 47, no. 3, pp. 261โ273, 2008.
[9] S. Singh and P. K. Jain, โFinite integral transform method tosolve asymmetric heat conduction in a multilayer annulus withtime-dependent boundary conditions,โ Nuclear Engineeringand Design, vol. 241, no. 1, pp. 144โ154, 2011.
[10] P. K. Jain, S. Singh, and Rizwan-uddin, โAnalytical solution totransient asymmetric heat conduction in a multilayer annulus,โJournal of Heat Transfer, vol. 131, no. 1, pp. 1โ7, 2009.
[11] P. K. Jain and S. Singh, โAn exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical
coordinates,โ International Journal of Heat and Mass Transfer,vol. 53, no. 9-10, pp. 2133โ2142, 2010.
[12] N. Dalir and S. S. Nourazar, โAnalytical solution of theproblem on the three-dimensional transient heat conductionin a multilayer cylinder,โ Journal of Engineering Physics andThermophysics, vol. 87, no. 1, pp. 89โ97, 2014.
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