International Journal of Thermophysics

, Volume 27, Issue 2, pp 596–613

# 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

## References

1. 1.
Fujimoto J.G., Liu J.M., and Ippen E.P. (1984). Phys. Rev 53:416Google Scholar
2. 2.
Brorson S.D., Fujimoto J.G., and Ippen E.P. (1987). Phys. Rev. Lett 59:1962
3. 3.
Elsayed-Ali H.E. (1991). Phys. Rev. B43:4488
4. 4.
Kaganov M.I., Lifshitz I.M., and Tanatarov L.V. (1957). Sov Phys JETP 4:173
5. 5.
Anisimov S.I., Kapeliovich B.L., and Perel’man T.L. (1974). Sov. Phys. JETP 39:375
6. 6.
Elsayed-Ali H.E., Norris T.B., Pessot M.A., and Mourou G.A. (1987). Phys. Rev. Lett 58:1212
7. 7.
Corkum P.B., Brunel F., and Sherman N.K. (1987). Phys. Rev. Lett. 61:2886
8. 8.
Qiu T.Q., and Tien C.L. (1992). Int. J. Heat Mass Transfer 35:719
9. 9.
Qiu T.Q., and Tien C.L. (1994). Int. J. Heat Mass Transfer 37:2789
10. 10.
Kiwan S. and Al-Nimr M.A. (2000). Jpn. J. Appl. Phys 39:4245
11. 11.
Al-Nimr M.A., Hader M., and Naji M. (2003). Int. J. Heat Mass Transfer 46:333
12. 12.
Qiu T.Q., and Tien C.L. (1993). J. Heat Transfer 115:835
13. 13.
Tzou D.Y. (1995). Int. J. Heat Mass Transfer 38:3231
14. 14.
Al-Nimr M.A., and Arpaci V.S. (2000). Int. J. Heat Mass Transfer 43:2021
15. 15.
Al-Nimr M.A.,, Haddad O.M., and Arpaci V.S. (1999). Heat Mass Transfer 35:459
16. 16.
Naji M., Al-Nimr M.A.,, and Hader M. (2003). Int J Thermophys. 24:545
17. 17.
Al-Nimr M.A., and Alkam M.K. (2003). Int. J. Thermophys. 24:577
18. 18.
Tzou D.Y. (1995). ASME J. Heat Transfer 117:8
19. 19.
Tzou D.Y. (1997). Macro- to Microscale Heat Transfer: The Lagging Behavior. Taylor & Francis, Washington, DC, Chap. 10.Google Scholar
20. 20.
Tzou D.Y. (2003). ASME J. Dynamic Systems, Measurement, and Control 125:563
21. 21.
Shih T.M. (1984). Numerical Heat Transfer. Hemisphere, New York, p. 55.
22. 22.
Chen H.T., and Liu K.C. (2003). Int. J. Heat Mass Transfer 46:2809
23. 23.
Chen H.T., and Liu K.C. (2003). Comput. Phys. Commun 150:31
24. 24.
Chen H.T., and Liu K.C. (2003). Int. J. Numer. Methods Eng. 57:637
25. 25.
Honig G., and Hirdes U. (1984). J. Comput. Appl. Math. 10:113
26. 26.
Hays-Stang K.J., Haji-Sheikh A. (1999). Int. J. Heat Mass Transfer 42:455