Heat flux formulation for 1D dual-phase lag equation

Abstract. In the paper a description of heat transfer in one-dimensional crystalline solids is presented. The lattice Boltzmann method based on Boltzmann transport equation is used to simulate the nanoscale heat transport in thin metal films. The coupled lattice Boltzmann equations for electrons and phonons are applied to analyze the heating process of thin metal films via laser pulse. Such approach in which the parameters appearing in the problem analyzed are treated as constant values is widely used, but in the paper the interval values of relaxation times and electron-phonon coupling factor are taken into account. The problem formulated has been solved by means of the interval lattice Boltzmann method using the rules of directed interval arithmetic. In the final part of the paper the results of numerical computations are shown. are still a subject of discussion [18]. The analogical problem is with the electron-phonon coupling factor. In the literature we can find a wide range of this value [19]. So it seems natural to take the interval values of relaxation times and coupling factor and this assumption is closer to the real physical conditions of the process analyzed.In the article the authors present an innovative approach of the described problem using the interval lattice Boltzmann method. Using interval numbers, the mathematical model reflects better the course of the heat flow. There was no such an approach in available literature until now. Moreover, there isn't any known commercial software using the interval LBM, although the presented method can easily be programmed.Here, the interval lattice Boltzmann method (ILBM) is presented with the approach of the directed interval arithmetic [16,20,21]. In this arithmetic a set of proper intervals is extended by improper intervals, and all arithmetic operations and functions are also extended. The main advantage of the directed interval arithmetic [20,22] upon the usual interval arithmetic [23] is that the obtained temperature intervals are much narrower and their width does not increase in time.In theory as well as in practice it is valuable to develop the interval version of the LBM. Directed interval arithmeticLet us consider a directed interval ā which can be defined as a set D of all directed pairs of real numbers defined as follows [17,20,24] where a − and a + denote the beginning and the end of the interval, respectively.

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“…The Boltzmann transport equations with the interval values of the relaxation time and the phonon-electron coupling [6] [7]…”

Section: Discussionmentioning

confidence: 99%

“…This kind of phenomena can be described by the Boltzmann transport equation (BTE) [2][3][4][5] but the other approaches can be also applied, e.g. [6][7][8]. The BTE is difficult to solve and for this reason in numerical computations the lattice Boltzmann method (LBM) is used [9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Modelling of transient heat transport in metal films using the interval lattice Boltzmann method

Piasecka-Belkhayat

Korczak

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…To solve both the basic problem and the additional one, the finite difference method in the version presented in [21] is applied. The geometrical mesh with constant step h and time step ∆t are introduced.…”

Section: Methods Of Solutionmentioning

confidence: 99%

“…Because the explicit scheme of the finite difference method is applied, the stability criteria should be fulfilled [21] ( ) The calculations were made using the explicit scheme of the finite difference method assuming the grid step h = 0.0002 m and time step ∆t = 0.01 s. In Figure 1 the distribution of sensitivity function after 60, 120 and 180 seconds is presented. Figures 2 and 3 illustrate the courses of temperature and sensitivity function at the points x = 0 (heated surface), x = 1 mm, x = 2 mm, x = 3 mm and x = 4 mm.…”

Section: Methods Of Solutionmentioning

confidence: 99%

1D generalized dual-phase lag equation. Sensitivity analysis with respect to the porosity

Kałuża

Majchrzak

Turchan

2016

J. Appl. Math. Comput. Mech.

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Abstract. Thermal processes occurring in the heated tissue are described by the 1D generalized dual-phase lag equation supplemented by appropriate boundary and initial conditions. Using the sensitivity analysis method, the additional problem connected with the porosity is formulated. Both problems are solved by means of the explicit scheme of the finite difference method. In this way it is possible to estimate the temperature changes due to the perturbation of porosity. In the final part of the paper, the example of computation is shown and the conclusions are formulated.

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“…The DPLE contains a second order time derivative and higher order mixed derivative in both time and space (e.g. [5][6][7][8] heat transfer analysis involves the use of two-temperature models. The two-temperature hyperbolic (parabolic) model consists of equations describing the temporal and spatial evolution of the lattice and electrons temperatures, the lattice and electron heat fluxes [9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Numerical modelling of the transient heat transport in a thin gold film using the fuzzy lattice Boltzmann method with α-cuts

Piasecka-Belkhayat

Korczak

2016

J. Appl. Math. Comput. Mech.

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Abstract. In this paper a description of heat transfer in one-dimensional crystalline solids is presented. The fuzzy lattice Boltzmann method based on the Boltzmann transport equation is used to simulate the nanoscale heat transport in thin metal films. The fuzzy coupled lattice Boltzmann equations for electrons and phonons are applied to analyze the heating process of thin metal films via a laser pulse. Such an approach in which the parameters appearing in the problem analyzed are treated as constant values is widely used. Here, the model with fuzzy values of relaxation times and an electron-phonon coupling factor is taken into account. The problem formulated has been solved by means of the fuzzy lattice Boltzmann method using the α-cuts and the rules of directed interval arithmetic. The application of α-cuts allows one to avoid complicated arithmetical operations in the fuzzy numbers set. In the final part of the paper the results of numerical computations are shown.

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Heat flux formulation for 1D dual-phase lag equation

Cited by 6 publications

References 5 publications

Modelling of transient heat transport in metal films using the interval lattice Boltzmann method

Modelling of transient heat transport in metal films using the interval lattice Boltzmann method

1D generalized dual-phase lag equation. Sensitivity analysis with respect to the porosity

Numerical modelling of the transient heat transport in a thin gold film using the fuzzy lattice Boltzmann method with α-cuts

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