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

  • Milad Mozafarifard
  • Aziz AzimiEmail author
  • Salem Mehrzad


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.


non-Fourier heat transfer fractional calculus inverse heat transfer Levenberg-Marquardt straight fin unsteady boundary condition 



Cross section of arbitrary profile/m2


Thermal wave speed/m•s−1


Specific heat capacity/J•kg−1•K−1


Generation number


Dimensionless convective heat transfer coefficient


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

i, j, n

Number of iterations


Sensitivity coefficient


Thermal conductivity/ W•m−1•K−1


Fin length/m


Unknown parameters in inverse problem


Perimeter of fin element/m


Heat flux/W•m−2


Internal heat generation/W•m−2


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


Least squares norm


Fin temperature/°C


Fin base temperature/°C


Average of fin base temperature/°C


Ambient temperature/°C

T0 Initial

temperature of fin/°C




Coordinate variable/m

GREEK Quantities


Fin thermal diffusivity/m2•s−1


Order of fractional derivative


Dimensionless temperature gradient relaxation time


Internal heat generation parameter/ K−1


G Dimensionless internal heat generation parameter


Dimensionless coordinate parameter


Dimensionless fin temperature


Dimensionless ambient temperature


Dimensionless heat flux relaxation time


Damping parameter


Dimensionless time




Heat flux relaxation time/s


Temperature gradient relaxation time/s


Amplitude of periodic base temperature


Dimensionless base temperature frequency


Base temperature frequency/s−1


Error tolerance


Weight factor of fractional derivative


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Copyright information

© Science Press, Institute of Engineering Thermophysics, CAS and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringShahid Chamran University of AhvazAhvazIran

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