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Research Papers: Conduction

Nonhomogeneous Dual-Phase-Lag Heat Conduction Problem: Analytical Solution and Select Case Studies

[+] Author and Article Information
Simon Julius

Turbomachinery and Heat Transfer Laboratory,
Faculty of Aerospace Engineering,
Technion—Israel Institute of Technology,
Haifa 3200003, Israel
e-mail: shimonjulius@tx.technion.ac.il

Boris Leizeronok

Mem. ASME
Turbomachinery and Heat Transfer Laboratory,
Faculty of Aerospace Engineering,
Technion—Israel Institute of Technology,
Haifa 3200003, Israel
e-mail: borisl@technion.ac.il

Beni Cukurel

Mem. ASME
Turbomachinery and Heat Transfer Laboratory,
Faculty of Aerospace Engineering,
Technion—Israel Institute of Technology,
Haifa 3200003, Israel
e-mail: beni@cukurel.org

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 27, 2016; final manuscript received July 16, 2017; published online October 10, 2017. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 140(3), 031301 (Oct 10, 2017) (22 pages) Paper No: HT-16-1694; doi: 10.1115/1.4037775 History: Received October 27, 2016; Revised July 16, 2017

Finite integral transform techniques are applied to solve the one-dimensional (1D) dual-phase heat conduction problem, and a comprehensive analysis is provided for general time-dependent heat generation and arbitrary combinations of various boundary conditions (Dirichlet, Neumann, and Robin). Through the dependence on the relative differences in heat flux and temperature relaxation times, this analytical solution effectively models both parabolic and hyperbolic heat conduction. In order to demonstrate several exemplary physical phenomena, four distinct cases that illustrate the wavelike heat conduction behavior are presented. In the first model, following an initial temperature spike in a slab, the thermal evolution portrays immediate dissipation in parabolic systems, whereas the dual-phase solution depicts wavelike temperature propagation—the intensity of which depends on the relaxation times. Next, the analysis of periodic surface heat flux at the slab boundaries provides evidence of interference patterns formed by temperature waves. In following, the study of Joule heating driven periodic generation inside the slab demonstrates that the steady-periodic parabolic temperature response depends on the ratio of pulsatile electrical excitation and the electrical resistivity of the slab. As for the dual-phase model, thermal resonance conditions are observed at distinct excitation frequencies. Building on findings of the other models, the case of moving constant-amplitude heat generation is considered, and the occurrences of thermal shock and thermal expansion waves are demonstrated at particular conditions.

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Figures

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Fig. 1

Relative contribution of the first 32 transcendental roots under representative Robin boundary conditions

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Fig. 2

Parabolic solution to spike initial condition with ϵ=10−7—temperature distribution for τq=τT=0

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Fig. 3

Solutions with parametric variation of τq and τT for invariant spike initial condition with ϵ=10−7

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Fig. 4

Solutions with parametric variation of spike initial condition distribution for invariant τq=10−9  s  and τT=10 −15 s

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Fig. 5

Parabolic and dual-phase solutions (3D view and x/L−t⋅f plane view) with parametric variation of periodic surface heat flux fluctuation frequency for invariant τq=10−4 sand τT=10 −8  s

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Fig. 6

Dual-phase solutions (3D view and x/L−t⋅f plane view) with parametric variation of τq and τT for invariant periodic surface heat flux fluctuation frequency ω=2π×10 kHz

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Fig. 7

Dual-phase solutions at slab centerpoint (x/L=0.5) with parametric variation of τq for invariant periodic surface heat flux fluctuation frequency ω=2π×10 kHz

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Fig. 8

Dual-phase solutions at slab boundary (x/L=0) with parametric variation of τT for invariant periodic surface heat flux fluctuation frequency ω=2π×10 kHz

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Fig. 9

Generation distributions, parabolic, and dual-phase solutions with parametric variation of μσ for invariant τq=10−6  sand τT=10 −8  s at frequency of ω=2π×10 kHz

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Fig. 10

Representative thermal Womersley number profiles

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Fig. 11

Generation Distribution for μσ=1000 s/m2 at frequency of ω=2π×10 kHz

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Fig. 12

Solutions with parametric variation of τq and τT for μσ=1000  s/m2 at frequency of ω=2π×10  kHz

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Fig. 13

Peak temperature amplitude of dual-phase solution with parametric variation of frequency and τq at invariant τT=10−8 s

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Fig. 14

Peak temperature amplitude of dual-phase solution with parametric variation of frequency and τT at invariant τq=2.5×10−4 s

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Fig. 15

Parabolic and dual-phase solutions with varying frequency for μσ=1000  s/m2

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Fig. 16

Solutions for moving generation source with parametric variation of U for invariant τq=1.28×10−11  s and τT=0  s

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Fig. 17

Solutions at slab centerpoint (x/L=0.5) for moving generation source with parametric variation of U at critical conditions (satisfied by changing τq)

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