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Numerical Heat Transfer, Part A: Applications
An International Journal of Computation and Methodology
Volume 73, 2018 - Issue 2
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Original Articles

Integral transform solution of integro-differential equations in conduction-radiation problems

ORCID Icon, ORCID Icon & ORCID Icon
Pages 94-114 | Received 23 Sep 2017, Accepted 20 Dec 2017, Published online: 23 Jan 2018
 

ABSTRACT

A new approach for solving nonlinear integro-differential equations in conductive-radiative heat transfer has been developed. The method relies on eigenfunctions expansions for the unknown potentials, following the hybrid analytical-numerical framework provided by the generalized integral transform technique. The problem of conjugated conduction–radiation in a finned-tube radiator is selected for illustrating the method, and a traditional numerical solution of the problem is performed for comparing the proposed approach. A thorough error analysis demonstrates that the proposed scheme is very effective for handling integro-differential problems. Finally, a parametric analysis is provided, demonstrating the effects of the dimensionless groups in the temperature distribution.

Nomenclature

𝒜s=

surface area

A, B, C, D=

integral coefficients

F=

configuration factor

h=

height

i, j=

summation indices

k=

thermal conductivity

L=

length

N=

norm

=

radiation–conduction parameter

r=

radius

t=

thickness

T=

temperature

W=

width

x, y=

coordinates components

X, Y=

dimensionless coordinates components

Greek symbols=
α1, α2=

constants

β=

configuration parameter

δ=

thickness

δij=

kronecker delta

θ=

homogenized dimensionless temperature

Θ=

dimensionless temperature

κ=

aspect ratio

λ=

configuration parameter

μ=

eigenvalues

σ=

Stefan–Boltzmann constant

ϕ=

filter

φ=

angle

Ψ=

eigenfunctions

ω=

ratio between length and width

Subscripts=
b=

base

c=

conduction

max=

maximum

r=

radiation

Superscripts=
=

dimensionless quantity

Overscripts=
¯=

transformed quantity

Nomenclature

𝒜s=

surface area

A, B, C, D=

integral coefficients

F=

configuration factor

h=

height

i, j=

summation indices

k=

thermal conductivity

L=

length

N=

norm

=

radiation–conduction parameter

r=

radius

t=

thickness

T=

temperature

W=

width

x, y=

coordinates components

X, Y=

dimensionless coordinates components

Greek symbols=
α1, α2=

constants

β=

configuration parameter

δ=

thickness

δij=

kronecker delta

θ=

homogenized dimensionless temperature

Θ=

dimensionless temperature

κ=

aspect ratio

λ=

configuration parameter

μ=

eigenvalues

σ=

Stefan–Boltzmann constant

ϕ=

filter

φ=

angle

Ψ=

eigenfunctions

ω=

ratio between length and width

Subscripts=
b=

base

c=

conduction

max=

maximum

r=

radiation

Superscripts=
=

dimensionless quantity

Overscripts=
¯=

transformed quantity

Additional information

Funding

The authors would like to acknowledge the financial support provided by the Brazilian Government Funding Agencies, CAPES, CNPq, and FAPERJ. This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grant Number 310458/2015-3], Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grant Number 477590/2013-6], Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro [Grant Number 111.068/2013] and, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro [Grant Number 102.216/2013].

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