Research Article Exact Analytical Solution for 3D Time ...three-dimensional 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 Β© 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 includeGreenβs 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 OΜzisΜ§ik [2], usingthe orthogonal expansion approach, solved the problem oftransient three-dimensional heat conduction in a compositeCartesian medium. Haji-Sheikh and Beck [3] used Greenβsfunction 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 π = constant and π = constant. However,nonhomogeneous boundary conditions of the third kind(convection) [11] are used in the π-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 (π-π-πcoordinates) or part-sphericalmultilayer geometries. Second,volumetric internal heat sources of nonuniform and time-dependent (π, π, π and π‘-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 π and π-directions, only the firstor second kind of homogeneous boundary condition can beapplied.
2. Mathematical Formulation
A π-layer composite spherical slab (π0β€ π β€ π
π, 0 β€ π β€
π, and 0 β€ π β€ π) is considered. All the layers have perfectthermal contact and are presumed to be isotropic in thermalproperties. πΌ
πand π
πare the temperature independent ther-
mal diffusivity and thermal conductivity of the πth layer. Atπ‘ = 0, the πth layer is at a specified temperature π
π(π, π, π)
and time dependent heat sources ππ(π, π, π, π‘) are switched
on in each radial layer. For π‘ > 0, homogenous boundaryconditions of first or second kind are applied to the angularsurfaces of π = 0 and π = π and azimuthal surfaces of π = 0and π = π. For the inner (π = 1, π = π
0) and the outer
(π = π, π = ππ) 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:
π2ππ
ππ2+2
π
πππ
ππ+1
π2π2ππ
ππ2+cot ππ2
πππ
ππ+
1
π2sin2ππ2ππ
ππ2
+ππ(π, π, π, π‘)
ππ
=1
πΌπ
πππ
ππ‘,
ππ= ππ(π, π, π, π‘) , π
0β€ π β€ π
π,
ππβ1β€ π β€ π
π, 1 β€ π β€ π,
0 β€ π β€ π, π < π,
0 β€ π β€ π, π < 2π,
π‘ β₯ 0.
(1)
The boundary conditions are as follows.
(i) Inner surface of 1st layer (π = 1):
π΄ inππ1
ππ(π0, π, π, π‘) + π΅inπ1 (π0, π, π, π‘) = πΆin. (2)
(ii) Outer surface of πth layer (π = π):
π΄outπππ
ππ(ππ, π, π, π‘) + π΅outππ (ππ, π, π, π‘) = πΆout. (3)
(iii) π = π surface (π = 1, 2, . . . , π):
ππ(π, π = π, π, π‘) = 0 or
πππ
ππ(π, π = π, π, π‘) = 0,
π = 1, . . . , π.
(4)
(iv) π = 0 surface (π = 1, 2, . . . , π):
ππ(π, π, π = 0, π‘) = 0 or
πππ
ππ(π, π, π = 0, π‘) = 0,
π = 1, . . . , π.
(5)
(v) π = π surface (π = 1, 2, . . . , π):
ππ(π, π, π = π, π‘) = 0 or
πππ
ππ(π, π, π = π, π‘) = 0,
π = 1, . . . , π.
(6)
(vi) Inner interface of the πth layer (π = 2, . . . , π):
ππ(ππβ1, π, π, π‘) = π
πβ1(ππβ1, π, π, π‘) π = 2, . . . , π,
ππ
πππ
ππ(ππβ1, π, π, π‘) = π
πβ1
πππβ1
ππ(ππβ1, π, π, π‘) π = 2, . . . , π.
(7)
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International Scholarly Research Notices 3
(vii) Outer interface of the πth layer (π = 1, . . . , π β 1):
ππ(ππ, π, π, π‘) = π
π+1(ππ, π, π, π‘) π = 1, . . . , π β 1, (8)
ππ
πππ
ππ(ππ, π, π, π‘) = π
π+1
πππ+1
ππ(ππ, π, π, π‘) π = 1, . . . , π β 1.
(9)
The initial condition is as follows:
ππ(π, π, π, π‘ = 0) = π
π(π, π, π) , 1 β€ π β€ π. (10)
It is worth mentioning that, at π = π0and π = π
π, 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 (π
0= 0) for
multilayered sphere ismodeled by assigning zero values toπ΅inand πΆin 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 (β2π = βπ2π),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:
β2ππ= βπ2ππβ
π2ππ
ππ2+2
π
πππ
ππ+1
π2π2ππ
ππ2
+cot ππ2
πππ
ππ+
1
π2sin2ππ2ππ
ππ2
= βπ2ππ.
(11)
Using the method of separation of variables, (11) is solved asfollows:
ππ(π, π, π) = π
π(π) Ξπ(π)Ξ¦π(π) , (12)
π πΞπΞ¦π
π πΞπΞ¦π
+2
π
π πΞπΞ¦π
π πΞπΞ¦π
+1
π2
π πΞπΞ¦π
π πΞπΞ¦π
+cot ππ2
π πΞπΞ¦π
π πΞπΞ¦π
+1
π2sin2ππ πΞπΞ¦π
π πΞπΞ¦π
= βπ2π πΞπΞ¦π
π πΞπΞ¦π
,
(13)
π π
π π
+2
π
π π
π π
+1
π2
Ξπ
Ξπ
+cot ππ2
Ξπ
Ξπ
+1
π2sin2πΞ¦π
Ξ¦π
= βπ2, (14)
π π
π π
+2
π
π π
π π
+1
π2
Ξπ
Ξπ
+cot ππ2
Ξπ
Ξπ
= β1
π2sin2πΞ¦π
Ξ¦π
β π2, (15)
sin2π(π2π π
π π
+ 2ππ π
π π
+Ξπ
Ξπ
+ cot πΞπ
Ξπ
+ π2π2)
= βΞ¦π
Ξ¦π
β = +]2,
(16)
Ξ¦
π+ ]2ππΞ¦π= 0 β Ξ¦
ππ(π) = π
1sin ]πππ + π2cos ]πππ,
π2π π
π π
+ 2ππ π
π π
+Ξπ
Ξπ
+ cot πΞπ
Ξπ
+ π2π2=
]2
sin2π,
(17)
β π2π π
π π
+ 2ππ π
π π
+ π2π2= β
Ξπ
Ξπ
β cot πΞπ
Ξπ
+]2
sin2π
= +π½2,
(18)
π2π
π+ 2ππ
π+ (π2
ππππ2β π½2
π) π π= 0
β π πππ(π) =
1
βπ[π3π½π½π+0.5
(πππππ)
+ π4ππ½π+0.5
(πππππ)] ,
Ξ
π+ (cot π)Ξ
π+ (π½2β
]2
sin2π)Ξπ= 0.
(19)
By using the following change of variable:
π = cos π β 1 β π2 = sin2π (20)the first and second derivatives of Ξ with respect to π areobtained as follows:
Ξ=πΞπ
ππ=πΞπ
ππ
ππ
ππ= β sin π
πΞπ
ππ,
Ξ
π=π2Ξπ
ππ2=π
ππ(πΞπ
ππ) =
π
ππ(β sin π
πΞπ
ππ)
= β cos ππΞπ
ππβ sin π π
ππ(πΞπ
ππ)
= sin2ππ2Ξπ
ππ2β cos π
πΞπ
ππ.
(21)
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4 International Scholarly Research Notices
Substituting (21) in (19) results in the following:
Ξ
π+ (cot π)Ξ
π+ (π½2β
]2
sin2π)Ξπ= 0
β sin2ππ2Ξπ
ππ2β cos π
πΞπ
ππ
+ (cos πsin π
)(β sin ππΞπ
ππ) + (π½
2β
]2
sin2π)Ξπ= 0
β (1 β π2)π2Ξπ
ππ2β 2π
πΞπ
ππ+ (π½2
πβ
]2ππ
1 β π2)Ξπ= 0.
(22)
If π½2π= π(π + 1), (22) is the associated Legendre equation. Its
solution is written as follows:
Ξππ(π) = π
5π]ππ
ππ(π) + π
6π
]ππ
ππ(π)
β Ξππ (π) = π5π
]ππ
ππ(cos π) + π6π
]ππ
ππ(cos π)
π]ππ
ππ(cos 0) =π]ππ
ππ(1) =ββπ
6= 0
β Ξππ(π) = π
]ππ
ππ(cos π) .
(23)
The problem eigenvalues are π2ππππ
= π2πππ+ ]2ππ. It should be
stated that the heat fluxes continuity at the interfaces of theradial layers gives the following:
Ξ¦ππ= Ξ¦π, ]
ππ= ]π,
ππππ= π1ππβ
πΌ1
πΌπ
.
(24)
The eigenfunctions π πππ(π), Ξ
ππ(π), and Ξ¦
ππ(π) in π-, π-, and
π-directions are derived by applying the boundary conditionsin each direction in the following equation:
π πππ(π) =
1
βπ[π3π½π(π + 1) + 0.5
(πππππ) + π4ππ(π + 1) + 0.5
(πππππ)] ,
Ξππ(π) = π
]π
ππ(cos π) ,
Ξ¦π(π) = π
1sin ]ππ + π2cos ]ππ.
(25)
It is assumed that the solution of the problem is in the formof a triple-series expansion of the derived eigenfunctions asfollows:
ππ(π, π, π, π‘) =
β
βπ=1
β
βπ = 1
β
βπ= 1
πππππ (π‘) π πππ (π) Ξππ (π)Ξ¦π (π) .
(26)The heat source term is also written as a triple-series expan-sion of the eigenfunctions such that
ππ(π, π, π, π‘) =
β
βπ=1
β
βπ = 1
β
βπ= 1
πππππ
(π‘) π πππ(π) Ξππ(π)Ξ¦π(π) ,
(27)
where the coefficient πππππ
(π‘) is obtained by the use of theorthogonality property as follows:
πππππ
(π‘) = (β«π
0
β«π
0
β«ππ
ππ β 1
ππ(π, π, π, π‘) π
2
Γ π πππ(π) Ξππ(π)Ξ¦π(π) ππ ππ ππ)
Γ (β«π
0
β«π
0
β«ππ
ππ β 1
π2π 2
πππ(π) Ξ2
ππ(π)
Γ Ξ¦2
π(π) ππ ππ ππ)
β1
.
(28)
Substitution of (26) and (27) in (1) results in the following:
πππππ
(π‘) π
πππ(π) Ξππ(π)Ξ¦ππ(π) +
2
ππππππ
(π‘) π
πππ(π) Ξππ(π)Ξ¦ππ(π) +
1
π2πππππ
(π‘) π πππ(π) Ξ
ππ(π)Ξ¦ππ(π)
+cot ππ2
πππππ
(π‘) π πππ(π) Ξ
ππ(π)Ξ¦ππ(π) +
1
π2sin2ππππππ
(π‘) π πππ(π) Ξππ(π)Ξ¦
ππ(π)
+1
ππ
πππππ
(π‘) π πππ(π) Ξππ(π)Ξ¦ππ(π) =
1
πΌπ
π
ππππ(π‘) π πππ(π) Ξππ(π)Ξ¦ππ(π) ,
(29)
βππππππ (π‘)
ππ‘+ (βπΌ
π)(
π πππ(π)
π πππ (π)
+2
π
π πππ(π)
π πππ (π)
+1
π2
Ξππ(π)
Ξππ (π)
+cot ππ2
Ξππ(π)
Ξππ (π)
+1
π2sin2πΞ¦ππ(π)
Ξ¦ππ(π)
)βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
= Ξππππ
πππππ
(π‘) =πΌπ
ππ
πππππ
(π‘) . (30)
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International Scholarly Research Notices 5
Equation (30) is a first-order nonhomogeneous ODE and hasthe following solution:
πππππ
(π‘) =πΌπ
ππ
πβΞπππππ‘β«π = π‘
π = 0
πππππ
(π) πΞπππππππ + π
π1πβΞπππππ‘.
(31)
Application of the initial condition (10) on (26) gives thefollowing:
ππ(π, π, π, π‘ = 0)
= ππ(π, π, π)
=
β
βπ=1
β
βπ = 1
β
βπ= 1
πππππ (0) π πππ (π) Ξππ (π)Ξ¦ππ (π) .
(32)
The coefficient ππ1of (31) is found by the use of the orthogo-
nality property for obtained eigenfunctions as follows:
ππ1
= πππππ (0)
=β«π
0β«π
0β«ππ
ππβ1
ππ(π, π, π) π2π
πππ(π) Ξππ(π)Ξ¦ππ(π) ππ ππ ππ
β«π
0β«π
0β«ππ
ππ β 1
π2π 2πππ(π) Ξ2ππ(π)Ξ¦2ππ(π) ππ ππ ππ
.
(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 β€ π β€ π3, 0 β€
π β€ π/2, and 0 β€ π β€ π) which is initially (π‘ = 0) at uniformunit temperature. For time π‘ > 0, thermal convection occurs,from the outer radial surface at π = π
3at (zero) ambient
temperature. However, the π = 0, π = π/2, π = 0, and π = πsurfaces are at uniformand constant zero temperatures.Theseboundary conditions lead to the following: π΄ in = 1, π΄out =π3, π΅in = 0, π΅out = β, πΆin = 0, and πΆout = 0. Additionally, the
uniformly distributed heat source ππ, π = 1, . . . , 3, is turned
on in each layer at π‘ = 0. The governing differential equationfor the 3D transient heat conduction with heat sources in thisthree-layer quarter-spherical region is as follows:
π2ππ
ππ2+2
π
πππ
ππ+1
π2π2ππ
ππ2+cot ππ2
πππ
ππ+
1
π2sin2ππ2ππ
ππ2+ππ
ππ
=1
πΌπ
πππ
ππ‘,
ππ= ππ(π, π, π, π‘) , 0 β€ π β€ π
3,
ππβ1β€ π β€ π
π, 1 β€ π β€ 3,
0 β€ π β€π
2,
0 β€ π β€ π,
π‘ β₯ 0.
(34)
The boundary conditions have the following forms:
ππ1
ππ(0, π, π, π‘) = 0, (35)
π3
ππ3
ππ(π3, π, π, π‘) + βπ
3(π3, π, π, π‘) = 0, (36)
ππ(π,
π
2, π, π‘) = 0, (37)
ππ(π, π, 0, π‘) = 0, (38)
ππ(π, π, π, π‘) = 0. (39)
(i) Inner interface of the πth layer (π = 2, 3):
ππ(ππ β 1, π, π, π‘) = π
π β 1(ππ β 1, π, π, π‘) ,
ππ
πππ
ππ(ππ β 1, π, π, π‘) = π
πβ1
πππ β 1
ππ(ππ β 1, π, π, π‘) .
(40)
(ii) Outer interface of the πth layer (π = 1, 2):
ππ(ππ, π, π, π‘) = π
π + 1(ππ, π, π, π‘) , (41)
ππ
πππ
ππ(ππ, π, π, π‘) = π
π + 1
πππ + 1
ππ(ππ, π, π, π‘) . (42)
The initial condition is as follows:
ππ(π, π, π, 0) = 1, 1 β€ π β€ 3. (43)
According to (25), by the use of the eigenfunction expansionmethod, the following solutions in π§-, π-, and π-directions areobtained from the associated eigenvalue problem:
Ξ¦π(π) = π
1sin ]ππ + π2cos ]ππ,
Ξππ(π) = π
]π
ππ(cos π) ,
π πππ (π) =
1
βπ[π3π½π(π + 1) + 0.5
(πππππ) + π4ππ(π + 1) + 0.5
(πππππ)] .
(44)
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6 International Scholarly Research Notices
The eigenvalues are π2ππππ
= π2πππ+]2ππ.The heat flux continuity
conditions at the interfaces imply the following:
ππππ= π1ππβ
πΌ1
πΌπ
. (45)
The eigenfunctionsπ πππ(π),Ξ
ππ(π), andΞ¦
π(π) in the π, π- and
π-directions are obtained by applying the relevant boundaryconditions in each direction. Application of the boundaryconditions in the π-direction due to (44) gives the following:
Ξ¦π(π) = π
1sin ]ππ + π2cos ]ππ
β
{{{{{{{
{{{{{{{
{
Ξ¦π (0) = 0 β π2 = 0
β Ξ¦π (π) = π1 sin ]ππ
Ξ¦π(π) = 0 β π
1sin ]ππ = 0
π1ΜΈ= 0
β sin ]ππ = 0 = sin (ππ) .
(46)
Then the π-direction eigenvalues and eigenfunction areobtained as follows:
]π= π, π = 1, 2, . . .
Ξ¦π(π) = sin (ππ) .
(47)
Using the π-direction boundary condition, (37), onΞ in (44)results in the following relation:
Ξππ(π) = π
π
ππ(cos π)
Ξππ(π/2) = 0
β ππ
ππ(cos(π
2)) = 0
β ππ
ππ(0) = 0,
(48)
where ππππ(0) = 0 is only satisfied when π are odd integers;
that is, π = 1, 3, 5, . . .. Thus the π-direction eigenvalues andthe eigenfunction are as follows:
π = 1, 3, 5, . . .
Ξππ(π) = π
π
ππ(cos π) .
(49)
Applying the π-direction boundary conditions, that is, (35)and (36), gives the following:
π πππ (π)
=1
βπ[π3π½π(π + 1) + 0.5
(πππππ) + π4ππ(π + 1) + 0.5
(πππππ)] ,
π = 1, 2, 3
β
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{
π (0) = 0 β π (0) = finite
ππ(π + 1) + 0.5
(0) =β
β π4= 0
β π πππ (π) = π3
1
βππ½π(π + 1) + 0.5
(πππππ)
ππ (π3) + βπ (π
3) = 0
β ππ3
1
βπ3π½π(π + 1) + 0.5
(πππππ3)
β ππ3
1
2π3βπ3
π½π(π + 1) + 0.5
(πππππ3)
+ βπ3
1
βπ3π½π(π + 1) + 0.5
(πππππ3) = 0
π3ΜΈ= 0
β1
βπ3[ππ½π(π + 1) + 0.5
(πππππ3)
+ (β βπ
2π3
) π½π(π + 1) + 0.5
(πππππ3)] = 0
β ππ½π(π + 1) + 0.5
(πππππ3)
+(β βπ
2π3
) π½π(π + 1) + 0.5
(πππππ3) = 0.
(50)
Thus the π-direction eigencondition and eigenfunction arederived as (]
π= π):
ππ½
π(π + 1) + 0.5(πππππ3) + (β β
π
2π3
) π½π(π + 1) + 0.5
(πππππ3) = 0,
π πππ (π) =
1
βππ½π(π + 1) + 0.5
(πππππ) .
(51)
According to (28) to (33), the coefficients πππππ
(π‘), ππ1, and
Ξππππ
are obtained as follows:
πππππ (π‘) = (β«
π
0
β«π
0
β«ππ
ππ β 1
πππ2 1
βππ½π½π+ 0.5
(πππππ) ππ
ππ(π)
Γ sin (ππ) ππ ππ ππ)
Γ (β«π
0
β«π
0
β«ππ
ππβ1
ππ½2
π½π+ 0.5
(πππππ) (ππ
ππ(π))2
Γ sin2 (ππ) ππ ππ ππ)β1
-
International Scholarly Research Notices 7
= (ππ(β«ππ
ππβ1
π3/2π½π½π+ 0.5
(πππππ) ππ)(β«
π
0
ππ
ππ(π) ππ)
Γ(β«π
0
sin (ππ) ππ))
Γ ((β«ππ
ππβ1
ππ½2
π½π+ 0.5
(πππππ) ππ) (β«
π
0
(ππ
ππ(π))2ππ)
Γ (β«π
0
sin2 (ππ) ππ))β1
= (ππ[(πππππ) π½π+1
(πππππ)]π = ππ
π = ππ β 1
Γ (β1
πcos (ππ))
π =π
π= 0
(β1
πcos (ππ))
π=π
π= 0
)
Γ ([(πππππ)2
2(π½2
π(πππππ) + π½
2
π+1(πππππ))]
π = ππ
π = ππ β 1
Γπ
2ΓπΏ
2)
β1
= (ππ[(πππππ) π½π+1
(πππππ)]π = ππ
π = ππ β 1
Γ1
π(1 β (β1)
π)
ΓπΏ
ππ(1 β (β1)
π))
Γ ([(πππππ)2
2(π½2
π(πππππ) + π½
2
π+1(πππππ))]
π = ππ
π = ππ β 1
ΓππΏ
4)
β1
β πππππ
(π‘) = πππππ
= ππ1ππ,
Ξππππ
= (βπΌπ)
Γ (π πππ(π)
π πππ(π)
+1
π
π πππ(π)
π πππ(π)
+1
π2
Ξππ(π)
Ξππ(π)
+Ξ¦ππ(π)
Ξ¦ππ(π))
= βπΌπ(π½π(πππππ)
π½π(πππππ)+1
π
π½π(πππππ)
π½π(πππππ)βπ2
π2β π2) .
(52)
Using the form of a triple-series expansion of the obtainedeigenfunctions, that is, (26), the solution is as follows:
ππ(π, π, π, π‘) =
β
βπ=1
β
βπ = 1
β
βπ= 1
πππππ (π‘)
1
βππ½π½π+ 0.5
Γ (πππππ) ππ
ππ(π) sin (ππ) ,
(53)
where the coefficient πππππ
(π‘) is attained as follows:
πππππ (π‘) =
πΌπ
ππ
πβΞπππππ‘πππππ
1
Ξππππ
(πΞπππππ‘β 1) + π
π1πβΞπππππ‘
=πΌππππππ
ππΞππππ
+ (1
ππ
βπΌπ
ππΞππππ
)πππππ
πβΞπππππ‘.
(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
π΄ in, π΅in, πΆin: Coefficients in (2)π΄out, π΅out, πΆout: Coefficients in (3)ππ(π, π, π): Initial temperature distribution in the
πth layer at π‘ = 0ππ(π, π, π, π‘): Volumetric heat source distribution in
the πth layerπππππ
: Coefficient in series expansion for heatsource (Equation (27))
β: Outer surface heat transfer coefficientπ½]π: Bessel function of the first kind of order
]π
ππ: Thermal conductivity of the πth layer
π: Radial coordinateππ: Outer radius for the πth layerπ πππ(π): Radial eigenfunctions for the πth layer
π‘: Timeππ(π, π, π, π‘): Temperature distribution for the πth
layerπππππ
: Coefficient in general solution(Equation (26)) dependent on initialcondition
π]π: Bessel function of the second kind oforder ]
π.
Greek Symbols
πΌπ: Thermal diffusivity of the πth layer
π: Azimuthal coordinateΞ¦ππ(π): Eigenfunctions in the π-direction
-
8 International Scholarly Research Notices
Ξππ(π): Eigenfunctions in the angular direction
ππππ: Radial eigenvalues
]π: Eigenvalues in the π-direction
π: Angle subtended by the multilayers inthe π-direction
π: Angle subtended by the multilayers inthe π-direction
Ξππππ
: Coefficient in (30).
Subscripts and Superscripts
π: 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
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[8] S. Singh, P. K. Jain, and Rizwan-uddin, βAnalytical solution totransient heat conduction in polar coordinates with multiplelayers in radial direction,β International Journal of ThermalSciences, vol. 47, no. 3, pp. 261β273, 2008.
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