In the present work, the heat flux and high order fluxes as well as their time derivatives are taken as independent variables and a new type of extended irreversible thermodynamics is developed. In the framework of this extended irreversible thermodynamics, the size dependence of the effective conductivity is investigated and a generalized single phase lagging heat conduction model including the size effect is established. Theoretically, it covers the diffusive to ballistic regime of heat conduction. The comparison with the experimental and theoretical results of silicon nanowires and thin films shows a good agreement in nano-scale regime.

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