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Numerical Simulation of Dual-Phase-Lag Model and Inverse Fractional Single-Phase-Lag Problem for the Non-Fourier Heat Conduction in a Straight Fin

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Abstract

In recent years, many studies have been done on heat transfer in the fin under unsteady boundary conditions using Fourier and non-Fourier models. In this paper, unsteady non-Fourier heat transfer in a straight fin having an internal heat source under periodic temperature at the base was investigated by solving numerically Dual-Phase-Lag and Fractional Single-Phase-Lag models. In this way, the governing equations of these models were presented for heat conduction analysis in the fin, and their results of the temperature distribution were validated using the theoretical results of Single and Dual-Phase-Lag models. After that, for the first time the order of fractional derivation and heat flux relaxation time of the fractional model were obtained for the straight fin problem under periodic temperature at the base using Levenberg-Marquardt parameter estimation method. To solve the inverse fractional heat conduction problem, the numerical results of Dual-Phase-Lag model were used as the inputs. The results obtained from Fractional Single-Phase-Lag model could predict the fin temperature distribution at unsteady boundary condition at the base as well as the Dual-Phase-Lag model could.

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Abbreviations

A :

Cross section of arbitrary profile/m2

C T :

Thermal wave speed/m•s−1

c :

Specific heat capacity/J•kg−1•K−1

G :

Generation number

H :

Dimensionless convective heat transfer coefficient

h :

Convective heat transfer coefficient/W•m−2•K−1

i, j, n :

Number of iterations

J :

Sensitivity coefficient

k :

Thermal conductivity/ W•m−1•K−1

L :

Fin length/m

P :

Unknown parameters in inverse problem

p :

Perimeter of fin element/m

q :

Heat flux/W•m−2

q gen :

Internal heat generation/W•m−2

q ∞ :

Internal heat generation at sink temperature/W•m−2

S :

Least squares norm

T :

Fin temperature/°C

T b :

Fin base temperature/°C

T b,m :

Average of fin base temperature/°C

T ∞ :

Ambient temperature/°C

T0 Initial :

temperature of fin/°C

t :

Time/s

X :

Coordinate variable/m

α :

Fin thermal diffusivity/m2•s−1

β :

Order of fractional derivative

Γ :

Dimensionless temperature gradient relaxation time

ε :

Internal heat generation parameter/ K−1

ε :

G Dimensionless internal heat generation parameter

η :

Dimensionless coordinate parameter

θ :

Dimensionless fin temperature

θ ∞ :

Dimensionless ambient temperature

Λ :

Dimensionless heat flux relaxation time

μ :

Damping parameter

ξ :

Dimensionless time

ρ :

Density/kg•m−3

τ q :

Heat flux relaxation time/s

τ T :

Temperature gradient relaxation time/s

φ :

Amplitude of periodic base temperature

ψ :

Dimensionless base temperature frequency

ω :

Base temperature frequency/s−1

ϵ :

Error tolerance

ω :

Weight factor of fractional derivative

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Authors and Affiliations

  1. Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, 6135785311, Iran

    Milad Mozafarifard, Aziz Azimi & Salem Mehrzad

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  1. Milad Mozafarifard
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  2. Aziz Azimi
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  3. Salem Mehrzad
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Correspondence to Aziz Azimi.

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Mozafarifard, M., Azimi, A. & Mehrzad, S. Numerical Simulation of Dual-Phase-Lag Model and Inverse Fractional Single-Phase-Lag Problem for the Non-Fourier Heat Conduction in a Straight Fin. J. Therm. Sci. 29, 632–646 (2020). https://doi.org/10.1007/s11630-019-1137-1

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  • DOI: https://doi.org/10.1007/s11630-019-1137-1

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