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Research Papers: Micro/Nanoscale Heat Transfer

Analytical Solution of the Hyperbolic Heat Conduction Equation for a Moving Finite Medium Under the Effect of Time Dependent Laser Heat Source

[+] Author and Article Information
R. T. Al-khairy

Department of Mathematics,
Applied Mathematics,
Dammam University,
Dammam 31113, Saudi Arabia
e-mail: ralkhairy@ud.edu.sa

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received March 9, 2012; final manuscript received July 4, 2012; published online October 8, 2012. Assoc. Editor: Robert D. Tzou.

J. Heat Transfer 134(12), 122402 (Oct 08, 2012) (7 pages) doi:10.1115/1.4007139 History: Received March 09, 2012; Revised July 04, 2012

This paper presents an analytical solution of the hyperbolic heat conduction equation for a moving finite medium under the effect of a time-dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by g(x,t) = I(t) (1 – R)μeμx while the finite body has an insulated boundary. The solution is obtained by the Laplace transforms method, and the discussion of solutions for two time characteristics of heat source capacities (instantaneous and exponential) is presented. The effect of the dimensionless medium velocity on the temperature profiles is examined in detail. It is found that there exists clear phase shifts in connection with the dimensionless velocity U in the spatial temperature distributions: the temperature curves with negative U values lag behind the reference curves with zero U, while the ones with positive U values precedes the reference curves. It is also found that the phase differences are the sole products of U, with increasing U predicting larger phase differences.

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References

Figures

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Fig. 1

Dimensionless temperature distributions for the instantaneous heat source; ϕ(η) = δ(η), β = 5, η = 0.1

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Fig. 8

Dimensionless temperature distributions for symmetrically heated slab with instantaneous heat source; ϕ(η) = δ(η), β = 5, U = 0

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Fig. 7

Dimensionless temperature distributions for the exponential heat source; ϕ(η) = exp(−νη), β = 1, U = 0.2, η = 0.4

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Fig. 6

Dimensionless temperature distributions for the exponential heat source; ϕ(η) = exp(−νη), β = 1, ν = 100, η = 0.1

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Fig. 5

Dimensionless temperature distributions for the instantaneous heat source; ϕ(η) = δ(η), η = 0.2, U = 0.4

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Fig. 4

Dimensionless temperature distributions for the instantaneous heat source at the middle of the slab; ϕ(η) = δ(η), β = 5, U = 0.1

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Fig. 3

Dimensionless temperature distributions for the instantaneous heat source; ϕ(η) = δ(η), β = 5, U = 0.1

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Fig. 2

Dimensionless temperature distributions for the instantaneous heat source; ϕ(η) = δ(η), β = 5, η = 0.3

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