  Starting from a classical thermodynamic approach, we derive rate-type
equations to describe the behavior of heat flow in deformable media.
Constitutive equations are defined in the material (Lagrangian) description
where the standard time derivative satisfies the principle of objectivity. The
statement of the Second Law is formulated in the classical form and the
thermodynamic restrictions are then developed following the Coleman-Noll
procedure. However, instead of the Clausius Duhem inequality we consider the
corresponding equality where the entropy production rate is prescribed by a
non-negative constitutive function. Both the free energy and the entropy
production are assumed to depend on a common set of independent variables
involving, in addition to temperature, both temperature gradient and heat-flux
vector together with their time derivatives. This approach results in rate-type
constitutive equations for the heat-flux vector that are intrinsically
consistent with the Second Law and easily amenable to analysis. In addition to
obtaining already known models (e.g. Cattaneo-Maxwell's, Jeffreys-like and
Green-Naghdi's heat conductors), this scheme allows us to build new and more
complex models of heat transport that may have applications in describing the
thermal behavior in nanosystems. Indeed, when higher order time derivatives of
the heat flux vector are involved, many different relaxation times occur within
the rate equation.
