Accurate numerical method for solving dual-phase-lagging equation with temperature jump boundary condition in nano heat conduction

Dai, Weizhong; Han, Fei; Sun, Zhi‐zhong

doi:10.1016/j.ijheatmasstransfer.2013.05.005

Cited by 30 publications

(9 citation statements)

References 32 publications

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“…The geometry of the solution is a nanoscale metal-oxide semiconductor field-effect transistor (MOS-FET). Recently, Dai et al [153] have employed an accurate high-order finite difference scheme for the numerical solution of the DPL model developed by [92,94] for 1-D and 2-D geometries.…”

Section: Finite Difference Methodsmentioning

confidence: 99%

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

2015

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The classical model of the Fourier's law is known as the most common constitutive relation for thermal transport in various engineering materials. Although the Fourier's law has been widely used in a variety of engineering application areas, there are many exceptional applications in which the Fourier's law is questionable. This paper gathers together such applications. Accordingly, the paper is divided into two parts. The first part reviews the papers pertaining to the fundamental theory of the phase-lagging models and the analytical and numerical solution approaches. The second part wrap ups the various applications of the phase-lagging models including the biological materials, ultra-high-speed laser heating, the problems involving moving media, micro/nanoscale heat transfer, multi-layered materials, the theory of thermoelasticity, heat transfer in the material defects, the diffusion problems we call as the non-Fick models, and some other applications. It is predicted that the interest in the field of phase-lagging heat transport has grown incredibly in recent years because they show good agreement with the experiments across a wide range of length and time scales.

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Section: Finite Difference Methodsmentioning

confidence: 99%

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

2015

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“…The convergence order in spatial direction is defined as

Order 1 = \log_{2} (\frac{e (2 h, τ)}{e (h, τ)})

for fixed sufficiently small τ . The convergence order in temporal direction is defined as

Order 2 = \log_{2} (\frac{e (h, 2 τ)}{e (h, τ)})

for fixed sufficiently small h .Example Consider the following initial and boundary value problem :

\frac{\partial T (x, y, t)}{\partial t} + \frac{\partial^{2} T (x, y, t)}{\partial t^{2}} = \frac{K_{n}^{2}}{3} Δ [B \frac{\partial T (x, y, t)}{\partial t} + T (x, y, t)], 0 < x, y < 1, t > 0,

where the initial condition is

T (x, y, 0) = x y + \cos (\frac{π}{2} x) \cos (\frac{π}{2} y), 0 \leq x, y \leq 1,

and the boundary conditions are

\begin{matrix} - α K_{n} \frac{\partial T}{\partial x} (0, \end{matrix}

…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…However, development of higher‐order accurate finite difference schemes for the DPL model with the temperature jump boundary condition is challenging due to the complex boundary condition. Recently, we have presented a higher‐order accurate and unconditionally stable compact finite difference scheme for solving 1D DPL equation with temperature jump boundary condition . In this article, we extend our study to a two‐dimensional (2D) case as follows:

\frac{\partial T}{\partial t} + \frac{\partial^{2} T}{\partial t^{2}} = \frac{K_{n}^{2}}{3} Δ (T + B \frac{\partial T}{\partial t}), (x, y) \in Ω, 0 \leq t \leq T,

\begin{matrix} {[- α K_{n} \frac{\partial T}{\partial x} + T] true|}_{x = 0} = φ (0, y, t), \\ {[α K_{n} \frac{\partial T}{\partial x} + T] true|}_{x = L_{1}} = φ (L_{1}, y, t), \end{matrix} 0 \leq y \leq L_{2}, 0 \leq t \leq T,

\begin{matrix} {[- α K_{n} \frac{\partial T}{\partial y} + T] true|}_{y = 0} = μ (x, 0, t), \\ {[α K_{n} \frac{\partial T}{\partial y} + T] true|}_{y = L_{2}} = μ \end{matrix}

…”

Section: Introductionmentioning

confidence: 98%

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Sun

Dai

et al. 2015

Numerical Methods Partial

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Dual-phase-lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher-order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher-order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher-order accurate and unconditionally stable compact finite difference scheme for solving one-dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two-dimensional case and develop a fourth-order accurate compact finite difference method in space coupled with the Crank-Nicolson method in time, where the Robin's boundary condition is approximated using a third-order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth-order in space and second-order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis.

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“…The Dual‐Phase‐Lagging (DPL) heat conduction equation, which was obtained based on the first‐order approximation of the non‐Fourier's law

q (x, t + τ_{q}) = - k \frac{\partial u}{\partial x} (x, t + τ_{u})

\frac{\partial u}{\partial t} + τ_{q} \frac{\partial^{2} u}{\partial t^{2}} = \frac{k}{C} (\frac{\partial^{2} u}{\partial x^{2}} + τ_{u} \frac{\partial^{3} u}{\partial t \partial x^{2}}) + \frac{1}{C} (Q + τ_{q} \frac{\partial Q}{\partial t}),

has been recently used to simulate heat transfer in nano‐structures . Here, k is the conductivity,

C = ϱ c_{p}

is the heat capacitance (in which ρ is the density and c p is the specific heat), Q is the heat source, τ q and τ T stand for the heat flux q and temperature gradient

\nabla u

phase lags, respectively.…”

Section: Introductionmentioning

confidence: 99%

A second‐order finite difference scheme for solving the dual‐phase‐lagging equation in a double‐layered nanoscale thin film

Sun

Dai

2016

Numerical Methods Partial

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This article considers the dual-phase-lagging (DPL) heat conduction equation in a double-layered nanoscale thin film with the temperature-jump boundary condition (i.e., Robin's boundary condition) and proposes a new thermal lagging effect interfacial condition between layers. A second-order accurate finite difference scheme for solving the heat conduction problem is then presented. In particular, at all inner grid points the scheme has the second-order temporal and spatial truncation errors, while at the boundary points and at the interfacial point the scheme has the second-order temporal truncation error and the first-order spatial truncation error. The obtained scheme is proved to be unconditionally stable and convergent, where the convergence order in L ∞ -norm is two in both space and time. A numerical example which has an exact solution is given to verify the accuracy of the scheme. The obtained scheme is finally applied to the thermal analysis for a gold layer on a chromium padding layer at nanoscale, which is irradiated by an ultrashort-pulsed laser.

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Accurate numerical method for solving dual-phase-lagging equation with temperature jump boundary condition in nano heat conduction

Cited by 30 publications

References 32 publications

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

A second‐order finite difference scheme for solving the dual‐phase‐lagging equation in a double‐layered nanoscale thin film

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