International Journal of Thermophysics

, Volume 33, Issue 3, pp 552–566 | Cite as

Investigation of 2D Transient Heat Transfer under the Effect of Dual-Phase-Lag Model in a Nanoscale Geometry

Article

Abstract

Analytical and numerical solutions of the 2D transient dual-phase-lag (DPL) heat conduction equation are presented in this article. The geometry is that of a simplified metal oxide semiconductor field effect transistor with a heater placed on it. A temperature jump boundary condition is used on all boundaries in order to consider boundary phonon scattering at the micro- and nanoscale. A combination of a Laplace transformation technique and separation of variables is used to solve governing equations analytically, and a three-level finite difference scheme is employed to generate numerical results. The results are illustrated for three Knudsen numbers of 0.1, 1, and 10 at different instants of time. It is seen that the wave characteristic of the DPL model is strengthened by increasing the Knudsen number. It is found that the combination of the DPL model with the proposed mixed-type temperature boundary condition has the potential to accurately predict a 2D temperature distribution not only within the transistor itself but also in the near-boundary region.

Keywords

DPL model Laplace transform Micro/nanoscale conduction MOSFET 

List of symbols

Variables

B

Phase-lag ratio

c

Specific heat (J·kg−1·°C−1)

FT

Laplace transform of temperature

k

Heat conduction coefficient (W·m−1·°C−1)

Kn

Knudsen number

Laplace transform operator

M

Normal function

n

Normal direction of boundaries

t

Time (s)

T

Temperature (K)

Th

Heater temperature (K)

Tw

Wall temperature (K)

Ts

Wall temperature jump (K)

T0

Ambient temperature (K)

x

Horizontal coordinate (m)

X

Separated function

y

Vertical coordinate (m)

Y

Separated function

Greeks

α

BC coefficient

\({\epsilon}\)

Inverse Laplace transform parameter

λ

Eigenvalues

Λ

Mean free path (m)

τ

Relaxation time (s)

Ω

Domain boundaries

Superscript

*

Non-dimensional condition

Subscripts

m, n

Counter

q

Heat flux

t

Temperature

w

Wall

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mechanical Engineering Department, Faculty of EngineeringUniversity of ZanjanZanjanIran
  2. 2.Mechanical Engineering DepartmentAmirkabir University of TechnologyTehranIran

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