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 |