HYPERBOLIC HEAT-CONDUCTION AND THE 2ND LAW

It is well known that Cattaneo's modification of the constitutive equation for heat flux yields a telegraph equation for determining temperature that models hyperbolic heat conduction with a finite thermal wave speed. By analyzing the solution of an example of thermal equilibration of an initially inhomogeneous state it is shown that Cattaneo's model violates a notion of the second law of thermodynamics because it predicts that heat may flow from cold to hot regions during finite time periods. In this paper we discuss modified restrictions associated with the second law and we propose an alternative formulation of the equations for heat conduction which retains the usual Fourier equation for heat flux but modifies the constitutive equations for internal energy, Helmholtz free energy, and entropy to include dependence on temperature rate. This alternative formulation satisfies the restrictions associated with the second law and produces the same telegraph equation for hyperbolic heat conduction as the Cattaneo model. However, since Fourier heat conduction is retained the new model predicts that heat always flows from hot to cold regions. An additional example of uniform heating is considered to show that the solution for temperature of the model proposed here differs from the solution associated with the parabolic and Cattanco models.