Improving convergence of summations in heat conduction

Beck, James V.; Cole, Kevin D.

doi:10.1016/j.ijheatmasstransfer.2006.06.032

Cited by 18 publications

(13 citation statements)

References 15 publications

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“…The series solutions available in some advanced heat conduction books for the temperature, and particularly the heat flux, converge slowly at the surface where a non-zero temperature is prescribed. Solutions with improved convergence are available [2,[4][5][6][7][8]. Even with these latter solutions, further improvement is possible based on a series identity given by Morse and Feshbach [9].…”

Section: Discussionmentioning

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%

“…See Ref. [8] for more discussion of this point. However, this recommendation is not usually the best one [6][7][8].…”

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%

“…Another method to obtain the steady-state component uses the complementary transient component [8] with the advantage of its exponential convergence. It is useful for locations away from the non-homogeneous surface since for sufficiently small dimensionless time it is the negative of the steady-state solution.…”

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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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…See Ref. [8] for more discussion of this point. However, this recommendation is not usually the best one [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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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Appendix B: Exact Solution X22B(y1pt1)0Y22B00T0

2023

Inverse Heat Conduction

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Improving convergence of summations in heat conduction

Cited by 18 publications

References 15 publications

Conduction in rectangular plates with boundary temperatures specified

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

Appendix B: Exact Solution X22B(y1pt1)0Y22B00T0

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