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International Journal of Thermophysics

, Volume 27, Issue 2, pp 596–613 | Cite as

Analysis for Microscopic Hyperbolic Two-Step Heat Transfer Problems

  • Kuo-Chi Liu
Article

This work analyzes the heat transfer problems in thin metal films using the microscopic hyperbolic two-step model. It is necessary in dealing with such problems to solve a set of the coupled energy equations or an equation containing higher-order mixed derivatives in both time and space. This present numerical scheme eliminates the coupling between energy equations with the Laplace transform technique and leads to a second-order governing differential equation in the transform domain. Afterward, the transformed second-order governing differential equation is discretized by the control volume scheme. To demonstrate the efficiency and accuracy of the present numerical scheme, a comparison between the present numerical results and the analytical solution is made. Theoretical insight into the hyperbolic two-step heat conduction is provided. Results show that the thermal propagation velocity is finite and is independent of the coupling factor and the volumetric heat capacity ratio between electrons and the lattice.

Keywords

ballistic behavior high-rate heating hybrid numerical scheme hyperbolic two-step model 

Nomenclature

C

volumetric heat capacity

E

dimensionless heating source, defined as \(E=\frac{S\tau }{C_{\rm e} T_{\rm r} } \widetilde {E}\)

\(\widetilde {E}\)

Laplace transform of E

G

coupling factor

k

thermal conductivity

distance between neighboring nodes

N

parameter, defined as \(N=\frac{G\tau }{C_{\rm e}}\)

Q

dimensionless heat flux, defined as \(Q=\frac{q}{T_{\rm r} \sqrt {kC_{\rm e}/\tau}}\)

\(\widetilde {Q}\)

Laplace transform of Q

q

heat flux

RC

parameter, defined as \(R_{\rm C} =\frac{C_{\rm l}}{C_{\rm e}}\)

S

heating source

s

Laplace transform parameter

T

temperature

T0

initial temperature

Tr

reference temperature

t

time

V

propagation speed of thermal signal, defined as \(V=\sqrt {k \mathord C_{\rm e} \tau } \)

x

space coordinate

Greek Symbols

 

β

parameter, defined in Eq. (9)

η

dimensionless space coordinate, defined as \(\eta\, =\,\frac{x}{\sqrt {k\tau /C_{\rm e}} }\)

θ

dimensionless temperature, defined as \(\theta =\frac{T-T_{\rm 0}}{T_{\rm r} }\)

\(\widetilde {\theta }\)

Laplace transform of θ

ξ

dimensionless time, defined as \(\xi =\frac{t}{\tau }\)

τ

relaxation time

Subscripts

 

e

electron

i

node number

l

lattice

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringFar East CollegeTaiwanRepublic of China

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