Fast-converging steady-state heat conduction in a rectangular parallelepiped

Crittenden, Paul E.; Cole, Kevin D.

doi:10.1016/s0017-9310(02)00066-2

Cited by 22 publications

(13 citation statements)

References 7 publications

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“…The derivation of the Fourier-space GF in Eq. (6) parallels that for steady-state G F given elsewhere [21]; however in the present work a is complex.…”

Section: Green's Functionsupporting

confidence: 79%

Analysis of Photothermal Characterization of Layered Materials ? Design of Optimal Experiments

Cole

2004

International Journal of Thermophysics

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In this paper numerical calculations are presented for the steady-periodic temperature in layered materials and functionally-graded materials to simulate photothermal methods for the measurement of thermal properties. No laboratory experiments were performed. The temperature is found from a new Green's function formulation which is particularly well-suited to machine calculation. The simulation method is verified by comparison with literature data for a layered material. The method is applied t o a class of two-component functionally-graded materials and results for temperature and sensitivity coefficients are presented. An optimality criterion, based on the sensitivity coefficients, is used for choosing what experimental conditions will be needed for photothermal measurements to determine the spatial distribution of thermal properties. This method for optimal experiment design is completely general and may be applied to any photothermal technique and to any functionally-graded material.KEY WORDS: functionally graded material; Green's functions; optimal experiment; photothermal; thermal properties I IntroductionFunctionally-graded (FG) materials are being studied as possible components of aerospace thermal protection systems. These materials include composites with epoxy and metal matrices, metal foams, or any structure with properties designed to vary with position. In the future when FG materials are specified as part of a vehicle program, part of the procurement process will involve certification that the material meets the specificat ions.To date there has been little research on accurate thermal characterization of FG materials. The present research is intended to close this gap in the procurement cycle by investigating photothermal methods for non-destructive and accurate measurement of thermal properties in FG materials. In this paper only numerical simulations are presented and no laboratory experiments were performed.Boulder, Colorado, U.S.A.

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“…The derivation of the Fourier-space GF in Eq. (6) parallels that for steady-state G F given elsewhere [21]; however in the present work a is complex.…”

Section: Green's Functionsupporting

confidence: 79%

Analysis of Photothermal Characterization of Layered Materials ? Design of Optimal Experiments

Cole

2004

International Journal of Thermophysics

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“…The complementary transient components have rapid exponential convergence, provided the dimensionless time is sufficiently large. However, exact conduction steady-state solutions for the temperature in a rectangle with prescribed temperature boundary conditions typically converge slowly at the surface with a non-homogeneous boundary condition [6]. The problem is acute for determining the heat flux at that surface [7].…”

Section: Introductionmentioning

confidence: 99%

“…Cole, Yen, and Crittenden [6,7] have clearly identified this problem and have proposed solutions. They have shown that there are multiple forms of the Green's function that may be used for 2D and 3D problems.…”

Section: Introductionmentioning

confidence: 99%

“…[8] for more discussion of this point. However, this recommendation is not usually the best one [6][7][8]. The convergence is usually improved by using eigenfunctions in a direction perpendicular to the nonhomogeneous surface [7].…”

Section: Introductionmentioning

confidence: 99%

“…As noted in [6][7][8], the series may contain a term that converges very poorly. These authors propose alternative Green's functions to improve convergence; the present paper removes this difficult term in a similar way.…”

Section: Introductionmentioning

confidence: 99%

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Conduction in rectangular plates with boundary temperatures specified

Beck

Wright

Haji‐Sheikh

et al. 2008

International Journal of Heat and Mass Transfer

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Steady-state components of heat conduction solutions may have very slowly convergent series for temperatures and non-convergent heat fluxes for temperature boundary conditions. Previous papers have proposed methods to remove these convergence problems. However, even more effective procedures based on insights of Morse and Feshbach are given herein. In some cases it is possible to replace poorly-convergent or non-convergent series by closed-form algebraic solutions. Examples are given.

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Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion

Zhou

Oldenburg

Rutqvist

et al. 2017

Water Resources Research

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There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1‐D, 2‐D, and 3‐D blocks. These infinite‐series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, td. In this paper, approximate solutions were developed by combining the error‐function‐series solutions for early times and the exponential‐series solutions for late times and by using time partitioning at the switchover time, td0. The combined solutions contain either the leading term of both series for normal‐accuracy approximations (with less than 0.003 relative error) or the first two terms for high‐accuracy approximations (with less than 10−7 relative error) for 1‐D isotropic (spheres, cylinders, slabs) and 2‐D/3‐D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1‐D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first‐order heat/mass flux for 2‐D/3‐D rectangular blocks were derived for normal‐accuracy approximations. These flux equations contain the early‐time solution with a three‐term polynomial in td and the late‐time solution with the limited‐term exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of low‐permeability blocks of varying shapes and sizes.

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Fast-converging steady-state heat conduction in a rectangular parallelepiped

Cited by 22 publications

References 7 publications

Analysis of Photothermal Characterization of Layered Materials ? Design of Optimal Experiments

Analysis of Photothermal Characterization of Layered Materials ? Design of Optimal Experiments

Conduction in rectangular plates with boundary temperatures specified

Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion

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