Analytical solution of non-Fourier heat conduction problem on a fin under periodic boundary conditions

Ahmadikia, Hossein; Rismanian, Milad

doi:10.1007/s12206-011-0720-5

Cited by 37 publications

(24 citation statements)

References 23 publications

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“…The evolution of θ b (x, t) is given by Equation (38), where C 1,2 are set by Equation (34). The evolution of the counterpart θ d (x, t) is governed by Equations (39) and (40).…”

Section: Exact Periodic Solutions To Ballistic Heat Transport In Thinmentioning

confidence: 99%

“…The r.h.s. of Equation (39) is set by Equation (38) for the ballistic component θ b (x, t), where C 1,2 are given by (34). The explicit solution θ d (x, t) to inhomogeneous DE (39) is possible, although it is much more cumbersome than in the Cauchy case:…”

Section: Exact Periodic Solutions To Ballistic Heat Transport In Thinmentioning

confidence: 99%

“…Solutions to Hyperbolic Heat Conduction Equation (HHCE) can be obtained both analytically and numerically [33][34][35][36][37][38][39][40], although numerical methods seem to be more commonly used [41][42][43][44]. Analytical study gives deeper insight in the problem; operational analytical approach and solutions to HHE were developed in [45][46][47][48][49][50].…”

Section: Introductionmentioning

confidence: 99%

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Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Zhukovsky

Oskolkov

Gubina

2018

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Abstract:One-dimensional equations of telegrapher's-type (TE) and Guyer-Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated-fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed.

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“…The evolution of θ b (x, t) is given by Equation (38), where C 1,2 are set by Equation (34). The evolution of the counterpart θ d (x, t) is governed by Equations (39) and (40).…”

Section: Exact Periodic Solutions To Ballistic Heat Transport In Thinmentioning

confidence: 99%

Section: Exact Periodic Solutions To Ballistic Heat Transport In Thinmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Zhukovsky

Oskolkov

Gubina

2018

Axioms

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“…This modified model can be used to solve the physically unreasonable deficiency of Fourier's law because it can capture the microscopic response in time [4,5]. A number of analytical and numerical studies in connection with the hyperbolic conduction heat transfer equation (CHTE) can be found in literature [6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, the Cattaneo-Vernotte (C-V) wave model in some cases may introduce exceptional behaviors [13,14] and negative thermal energies [15] and may violate the second law of thermodynamics [7,16]. To overcome these shortcomings, Tzou [17,18] introduced a dualphase-lag (DPL) model for heat conduction by lumping microstructural effects into delayed temporal responses in the macroscopic formulation, which has been supported experimentally [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Periodic Heat Transfer in Convective Fins Based on Dual-Phase-Lag Theory

Askarizadeh

Ahmadikia

2016

Journal of Thermophysics and Heat Transfer

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Heat transfer enhancement through extended surfaces is crucial in modern engineering applications such as microelectromechanical systems and electronic components. Conduction heat transfer is the only way to achieve this objective when the mixing augmentation is not possible. This paper investigates the effects of non-Fourier thermal conduction in convective straight fins with arbitrary constant cross section under periodic boundary conditions by introducing the exact analytical solution for the dual-phase-lag heat conduction. The corresponding analytical approach is developed through the Laplace transform method and inversion theorem. To help advance the understanding of fin thermophysical behavior, a generalized model of conduction heat transfer equation is used for studying all of the interpretations to Fourier-based model, namely, parabolic thermal diffusion, hyperbolic thermal wave, and dual-phase-lag models. Heat flux and temperature gradient relaxation times are the characteristics of the dual-phase-lag model, and the simulation results are strictly a function of these two parameters. Therefore, fin temperature distributions are presented for flux precedence and gradient precedence heat flow regimes. Calculations are performed to investigate the influence of temperature gradient relaxation time on the hyperbolic heat conduction characteristics of non-Fourier fins. Nomenclature A= fin cross section, m 2 C = thermal wave speed, m · s −1 C 1 -C 4 = constant coefficients c = specific heat of fin, J · kg −1 · K −1 dA s = peripheral surface of fin element, m 2 H = dimensionless convective heat transfer coefficient h = convective heat transfer coefficient, W · m −2 · K −1 k = fin thermal conductivity, W · m −1 · K −1 L = fin length, m n = nonnegative integer number p = perimeter of fin element, m q = heat transfer rate, W · m −2 r = nonnegative integer number s = Laplace domain parameter T = fin temperature,°C T b = fin base temperature,°C T b;m = average of fin base temperature,°C T 0 = initial temperature of fin,°C T ∞ = ambient temperature,°C t = time, s x = coordinate variable, m α = fin thermal diffusivity, m 2 · s −1 Γ = dimensionless temperature gradient relaxation time η = dimensionless coordinate parameter θ = dimensionless fin temperature θ ∞ = dimensionless ambient temperature Λ = dimensionless heat flux relaxation time λ n = 2n 1π∕2, eigenvalues ξ = dimensionless time ρ = fin material density, kg · m −3 τ q = heat flux relaxation time, s τ T = temperature gradient relaxation time, s φ = amplitude of periodic base temperature Ω = dimensionless base temperature frequency ω = base temperature frequency, s −1

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Thermal behaviour of convective‐radiative porous fins under periodic thermal conditions

2018

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In the present study, the effects of non-Fourier, convection, and radiation heat transfer are investigated in a porous fin under periodic thermal conditions. The porosity effect on the fin that allows the flow to infiltrate is formulated using Darcy's model. A nonlinear partial differential equation has been obtained by energy balance for the porous fin solved by a numerical method. The effects of buoyancy or natural convection parameter (N p ), the radiation parameter (N r ), the convection parameter (N c ), dimensionless relaxation time (C), and dimensionless frequency of the base temperature oscillation (v) on temperature distribution are studied. Increasing the values of C, as the non-Fourier condition of heat transfer, led to a discontinuity in the dimensionless temperature distribution with smaller values of h. The heat transfer rate of the porous fin has been increased by increasing N c , N r , and N p , of which the N r had the strongest effect on heat transfer in comparison with other parameters.

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Analytical solution of non-Fourier heat conduction problem on a fin under periodic boundary conditions

Cited by 37 publications

References 23 publications

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Periodic Heat Transfer in Convective Fins Based on Dual-Phase-Lag Theory

Thermal behaviour of convective‐radiative porous fins under periodic thermal conditions

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