Green's functions, temperature and heat flux in the rectangle

Cole, Kevin D.; Yen, David H. Y.

doi:10.1016/s0017-9310(01)00040-0

Cited by 33 publications

(30 citation statements)

References 3 publications

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“…The problem described by Eqs. (1)- (3) is investigated in [7] and substantial improvements are given compared to the solutions commonly given in textbooks. The methods herein present further improvement and even eliminate summations for certain aspect ratios and accuracies.…”

Section: Problemmentioning

confidence: 99%

“…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]. Solutions based on the separation of variables method, which we relate to the long cotime Green's function (with cotime defined as u = t À s), are particularly prone to this problem; the series for the heat flux normal to the surface may even fail to converge at that surface.…”

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%

See 4 more Smart Citations

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: Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Temperature prediction for multi‐dimensional domains in standard fire

Wang¹,

Tan²

2006

Commun. Numer. Meth. Engng.

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SUMMARYThe existing 1D analytical solution for heat conduction inside structural members plays an important role in the estimation of temperature field and prediction of member strength in fire. In this paper, the novelty lies in a set of proposed analytical formulations for the conduction heat transfer within multi-dimensional (2D and 3D) domains subjected to a particular time-varying boundary condition, i.e. the standard fire conditions. Solutions of multi-dimensional conduction are based on the eigenfunction approach using the technique of separation of variables. The multi-dimensional effect is incorporated by the product of respective solutions of 1D transient heat conduction (1D step functions) of the plate and the infinite cylinder that serve as a set of basis for derivation of analytical solutions. Fire boundary conditions are incorporated using the Duhamel's principle, where the time-varying fire temperature curve is imposed on the step functions. In this paper, the boundary conditions of the first and the third kind are considered, respectively. The analytical solutions are compared with results obtained from finite element analysis for verification.

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Green's functions, temperature and heat flux in the rectangle

Cited by 33 publications

References 3 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

Temperature prediction for multi‐dimensional domains in standard fire

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