Non-linear heat transfer of solids in orthogonal coordinate systems

Cobble, M.H.

doi:10.1016/0020-7462(67)90008-x

Cited by 16 publications

(13 citation statements)

References 4 publications

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“…3, (as well as in Figs. 4,5) confirm the comments that the accuracy of the integral-balance solutions (both HBIM and DIM) is better when β < 0. The range of variations of the pointwise errors is typical for the integral-balance solutions [1], i.e.…”

Section: Benchmarking Of the Approximate Solutions With Optimal Exponsupporting

confidence: 82%

“…It is worth noting, that in these solutions [1,23] the non-linear transform is mainly applied to the surface temperature (at x = 0). These solutions are not popular due to the inherent property of HBIM to predetermine the accuracy of approximation when a fixed order of the assumed profiles is used [1,3] as well as due to the difficulties emerging in the reversion of the solution in the terms of T, when the order of the polynomial approximation is high [4]. The recent applications of the simple HBIM [24] and the double-integral balance method (DIM) [25] to Eq.…”

Section: Problem Formulationmentioning

confidence: 98%

“…Equation (1b) and the relationship (1c) are related mainly to the temperature-dependent thermal conductivity k = k 0 (1 + βT ) assuming the product ρC p temperatureindependent [4][5][6][7].…”

Section: Problem Formulationmentioning

confidence: 99%

“…(1c) into a linear diffusion equation by applying (2a), namely Then, the solution of (2c) has been performed by, series expansion [4], separation of variables [19,20], homotopyperturbation method [22], etc. The final solution in term of T may be developed either analytically [4] by the series reversion method [21] or by numerical methods [22].…”

Section: Problem Formulationmentioning

confidence: 99%

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On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity

Fabre

Hristov

2016

Heat Mass Transfer

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relationship (Eqs. 9, 10) (m 2 /s) bCoefficient in Eq. (24b) which should be defined trough the initial condition δ (t = 0) = 0 C p Specific heat capacity (J/kg) E L (n, β, t) Squared-error function in accordance with the Langford criterion (Eq. 34) E LT (n, β, t) Squared-error function in accordance with the Langford criterion (Eq. 35) E Mq (p, β, t) Squared-error function in accordance with the Langford criterion and fixed flux BC problem (Eq. 76) e LT (n, β, t) Squared-error sub-function in accordance with the Langford criterion (Eq. 35) e Lq (p, β) Squared-error sub-function in accordance with the Langford criterion and the fixed flux BC problem k Thermal conductivity (W/mK) k (T) Temperature-dependent thermal conductivity (W/mK) k 0 Thermal conductivity of the linear problem (β = 0) (W/mK) m Dimensionless parameter of the nonlinearity (power-law diffusivity) n Dimensionless exponent of the parabolic profile p Dimensionless exponent of the parabolic profile of the assumed profile used to solve Eq. (48) (m) Abstract Closed form approximate solutions to nonlinear transient heat conduction with linearly temperaturedependent thermal diffusivity have been developed by the integral-balance integral method under transient conditions. The solutions uses improved direct approaches of the integral method and avoid the commonly used linearization by the Kirchhoff transformation. The main steps in the new solutions are improvements in the integration technique of the double-integration technique and the optimization of the exponent of the approximate parabolic profile with unspecified exponent. Solutions to Dirichlet and Neumann boundary condition problems have been developed as examples by the classical Heat-balance integral method (HBIM) and the Double-integration method (DIM). Additional examples with HBIM and DIM solutions to cases when the Kirchhoff transform is initially applied have been developed. List of symbolsThermal diffusivity (m 2 /s) a 0 Thermal diffusivity of the linear problem (β = 0) (m 2 /s) a p Thermal diffusivity coefficient in the case of power-law non-linear

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Section: Benchmarking Of the Approximate Solutions With Optimal Exponsupporting

confidence: 82%

Section: Problem Formulationmentioning

confidence: 98%

Section: Problem Formulationmentioning

confidence: 99%

Section: Problem Formulationmentioning

confidence: 99%

See 2 more Smart Citations

On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity

Fabre

Hristov

2016

Heat Mass Transfer

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show abstract

“…then the problem defined by equations (1), (2), (4) and the boundary and initial conditions can be solved analytically, see [2]. Let us call this problem and the original one as problems A and B or cases A and B respectively.…”

Section: Controlling Equationsmentioning

confidence: 99%