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Analysis for Microscopic Hyperbolic Two-Step Heat Transfer Problems

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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.

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Abbreviations

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

ℓ:

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

R C :

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

S :

heating source

s :

Laplace transform parameter

T :

temperature

T 0 :

initial temperature

T r :

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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Authors and Affiliations

  1. Department of Mechanical Engineering, Far East College, 49 Chung Hua Rd., Hsin-Shih Tainan Prefecture, Taiwan, 744, Republic of China

    Kuo-Chi Liu

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  1. Kuo-Chi Liu
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Correspondence to Kuo-Chi Liu.

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Liu, KC. Analysis for Microscopic Hyperbolic Two-Step Heat Transfer Problems. Int J Thermophys 27, 596–613 (2006). https://doi.org/10.1007/s10765-005-0006-1

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