Flux formulation of hyperbolic heat conduction

Frankel, J. I.; Vick, Brian; Özişik, M.N.

doi:10.1063/1.335795

Cited by 98 publications

(26 citation statements)

References 21 publications

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“…(2a) with the Green's function denoted by G(x, y, t/x 0 , y 0 , t 0 ) and integrating over the domains of interest. Here G(effect/cause) notation is used [31]. Doing so produces…”

Section: Derivation Of Integral Relationship: Green's Function Methodmentioning

confidence: 99%

A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

Frankel

Keyhani

Arimilli

et al. 2007

Z. angew. Math. Phys.

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This paper derives a new integral relationship between heat flux and temperature in a transient, two-dimensional heat conducting half space. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. The analytic expression provides the local heat flux perpendicular to the front surface solely based on an embedded line of temperature sensors parallel to the surface. The relationship does not require apriori knowledge of the surface boundary condition. A new sensor strategy is analytically conceived based on the integral relationship for estimating the local, in-depth heat flux without surface instrumentation. It should further be clarified that the integral relationship requires only knowledge of the local, in-depth temperature and heating/cooling rate (time rate of change of temperature). The resulting formulation is mildly ill-posed and either requires digital filtering of the temperature signal to remove high frequency components of noise or the development of direct heating/cooling rate sensors. This paper (a) develops the new mathematical relationship; (b) demonstrates that the proposed relationship reduces to well-known (i) one-dimensional results under the appropriate assumptions; and, (ii) two-dimensional surface results; and, (c) provides a simple numerical example validating the concept.

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“…(2a) with the Green's function denoted by G(x, y, t/x 0 , y 0 , t 0 ) and integrating over the domains of interest. Here G(effect/cause) notation is used [31]. Doing so produces…”

Section: Derivation Of Integral Relationship: Green's Function Methodmentioning

confidence: 99%

A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

Frankel

Keyhani

Arimilli

et al. 2007

Z. angew. Math. Phys.

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“…respectively, is highly useful and revealing in many applications [19]. It is possible to express the heat equation in heat flux, q (x, t) as…”

Section: Mathematical Formulationsmentioning

confidence: 99%

Regularization of inverse heat conduction by combination of rate sensor analysis and analytic continuation

Frankel

2006

J Eng Math

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This paper describes some recent observations associated with (1) clarifying and expanding upon the integral relationship between temperature and heat flux in a half-space; (2) offering an analyticcontinuation approach for estimating the surface temperature and heat flux in a one-dimensional geometry based on embedded measurements; and, (3) offering a novel digital filter that supports the use of analytic continuation based on a minimal number of embedded sensors. Key to future inverse analysis must be the proper understanding and generation of rate data associated with both the temperature and heat flux at the embedded location. For this paper, some results are presented that are theoretrically motivated but presently adapted to implement digital filtering. A pulsed surface heat flux is reconstructed by way of a single thermocouple sensor located at a well-defined embedded location in a half space. The proposed low-pass, Gaussian digital filter requires the specification of a cut-off frequency that is obtained by viewing the power spectra of the temperature signal as generated by the Discrete Fourier Transform (DFT). With this in hand, and through the use of an integral relationship between the local temperature and heat flux at the embedded location, the embedded heat flux can be accurately estimated. The time derivatives of the filtered temperature and heat flux are approximated by a simple finite-difference method to provide a sufficient number of terms required by the Taylor series for estimating (i.e., the projection) the surface temperature and heat flux. A numerical example demonstrates the accuracy of the proposed scheme. A series of appendices are offered that describe the mathematical details omitted in the body for ease of reading. These appendices contain important and subtle details germane to future studies.

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“…(4) for axisymmetric conduction in cylindrically bounded domains. Several solutions for finite media have been given in [9][10][11][12][13][14][15]. Tang and Araki [16,17] have solved the problem of non-Fourier heat conduction in a finite medium under periodic surface disturbance.…”

Section: Introductionmentioning

confidence: 99%

Axisymmetric non-Fourier temperature field in a hollow sphere

Moosaie

2008

Arch Appl Mech

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The non-Fourier axisymmetric (2+1)-dimensional temperature field within a hollow sphere is analytically investigated by the solution of the well-known Cattaneo-Vernotte hyperbolic heat conduction equation. The material is assumed to be homogeneous and isotropic with temperature-independent thermal properties. The method of solution is the standard separation of variables method. General linear time-independent boundary conditions are considered. Ultimately, the presented solution is applied to a (1+1)-as well as a (2+1)-dimensional problem, and their respective non-Fourier thermal behavior is studied. The present solution can be reduced to special cases of interest by choosing appropriate boundary conditions parameters.

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Flux formulation of hyperbolic heat conduction

Cited by 98 publications

References 21 publications

A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

Regularization of inverse heat conduction by combination of rate sensor analysis and analytic continuation

Axisymmetric non-Fourier temperature field in a hollow sphere

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