Hyperbolic model

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The classical heat conduction theory based on Fourier’s law assumes that thermal disturbance propagates with an infinite speed. As heat conduction is accomplished by successive collision of the energy carriers (phonons or electrons), the propagation of thermal disturbance is always at a finite speed. This is particularly important for those processes involving extremely short times, cryogenic temperatures, or high heat fluxes. To account for the finite propagation speed of thermal wave, the Cattaneo-Vernotte thermal wave model can be used

{\mathbf{q''}} + \tau \frac{{\partial {\mathbf{q''}}}}{{\partial t}} =  - k\nabla T    \qquad \qquad(1)

where τ is the relaxation time that can be interpreted as the time scale at which intrinsic length scale of thermal diffusion (\sqrt {\alpha t} ) is equal to the intrinsic length scale of thermal wave (ct) (Tzou and Li, 1993; Tzou, 1997), where c is the thermal propagation speed. Thus the relaxation time is

\tau  = \alpha /{c^2}    \qquad \qquad(2)

Substituting eq. (1) into the following energy equation:

\rho {c_p}\frac{{\partial T}}{{\partial t}} =  - \nabla  \cdot {\mathbf{q''}} + q'''

the following energy equation is obtained

\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} + \frac{\tau }{\alpha }\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {\nabla ^2}T + \frac{1}{k}\left( {q''' + \tau \frac{{\partial q'''}}{{\partial t}}} \right)    \qquad \qquad(3)

Since the second order derivative of temperature with respect to time appears in eq. (3), it is referred to as the hyperbolic heat conduction model. Mathematically, eq. (3) is a thermal wave equation and the thermal diffusivity appears as a dumping effect of thermal propagation. With appropriate relaxation times, eq. (3) can be used to describe temperatures of different energy carriers – such as phonon and electron temperatures – as will be discussed later.


Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Tzou, D. Y., 1997, Macro- to Microscale Heat Transfer, Taylor and Francis, Washington, DC.

Tzou, D.Y., and Li, J., 1993, “Thermal Wave Emanating from a Fast-Moving Heat Source with a Finite Dimension,” ASME Journal of Heat Transfer, Vol. 115, pp. 526-532.

Wang, G.X., and Prasad, V., 2000, “Microscale Heat and Mass Transfer and non-Equilibrium Phase Change in Rapid Solidification,” Materials Science and Engineering, A., Vol. 292, pp. 142-148.

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