As a paradigm for heat conduction in 1 dimension, we propose a class of
models represented by chains of identical cells, each one of which containing
an energy storage device called a "tank". Energy exchange among tanks is
mediated by tracer particles, which are injected at characteristic temperatures
and rates from heat baths at the two ends of the chain. For stochastic and
Hamiltonian models of this type, we develop a theory that allows one to derive
rigorously -- under physically natural assumptions -- macroscopic equations for
quantities related to heat transport, including mean energy profiles and tracer
densities. Concrete examples are treated for illustration, and the validity of
the Fourier Law in the present context is discussed.