Comparison of Solutions from Parabolic and Hyperbolic Models for Transient Heat Conduction in Semi-Infinite Medium

Abstract

The expression for the transient temperature during damped wave conduction and relaxation developed by Baumeister and Hamill by the method of Laplace transforms was further integrated. A Chebyshev polynomial approximation was used for the integrand with a modified Bessel composite function in space and time. A telescoping power series leads to a more useful expression for the transient temperature. By the method of relativistic transformation, the transient temperature during damped wave conduction and relaxation was developed. There are four regimes to the solution. These include: (i) a regime comprising a Bessel composite function in space and time, (ii) another regime comprising a modified Bessel composite function in space and time, (iii) the temperature solution at the wave front was also developed separately, and (iv) the fourth regime at a given location X in the medium is at times less than the inertial thermal lag time. In this regime, the temperature was found to be unchanged at the initial condition. The solution for the transient temperature from the method of relativistic transformation is compared side by side with the solution for the transient temperature from the method of Chebyshev economization. Both solutions are within 12 % of each other. For conditions close to the wave front, the solution from the Chebyshev economization is expected to be close to the exact solution and was found to be within 2 % of the solution from the method of relativistic transformation. Far from the wave front, i.e., close to the surface, the numerical error from the method of Chebyshev economization is expected to be significant and verified by a specific example. The solution for transient surface heat flux from the parabolic Fourier heart conduction model and the hyperbolic damped wave conduction and relaxation models are compared with each other. For τ > 1/2 the parabolic and hyperbolic solutions are within 10 % of each other. The parabolic model has a "blow-up" as τ → 0, and the hyperbolic model is devoid of singularities. The transient temperature from the Chebyshev economization is within an average of 25 % of the error function solution for the parabolic Fourier heat conduction model. A penetration distance beyond which there is no effect of the step change in the boundary is predicted using the relativistic transformation model. KeywordsDamped wave conduction and relaxation · Laplace transforms · Microscale heat transfer · Parabolic and hyperbolic models · Relativistic transformation Nomenclature erf (r ) Error function erf(r ) = 2 √ π r 0 e −r 2 dr H (τ ) Space integrated temperature I 0 (x) Modified Bessel function of the first kind and zeroth order I 1 (x) Modified Bessel function of the first kind and first order J 0 (x) Bessel function of the first kind and zeroth order k Thermal conductivity of semi-infinite medium (W · m −1 · K −1 ) K 0 (x) Modified Bessel function of the second kind and zeroth order mInitial temperature (K) T s Surface temperature (K) T n (r ) Chebyshev polynomial (Tables 1, 2) uBesse...

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