The phase-lag concept in the thermal behavior of lumped systems

The phase-lag concept in the wave theory of heat conduction is extended to describe the thermal behavior of lumped systems. It is assumed that a phase-lag exists between the convection heat flux from the lumped system and the temperature difference between the lumped system and its surroundings. It is found that the dimensionless delay time τ is an important parameter in specifying the qualitative behavior of the lumped system. The phase-lag concept has no significant effects on the thermal behavior of lumped systems having τ < 1/4. In this case, the classical theory without delay can give an accurate prediction for the system thermal behavior. On the other hand, the phase-lag concept changes the quantitative and qualitative behavior of systems having τ > 1/4 and these changes are enhanced as τ increases. However, it is shown that the thermal behavior of systems having τ > 1/4 violates the second law of thermodynamics. The physical reasoning for this violation is explained. Also, the phase-lag concept is extended to describe the thermal behavior of composite system which consists of two domains each is lumped at different temperature.