  The size-dependent and flux-dependent effective thermal conductivity of
narrow channels filled with He II is analyzed. The classical Landau evaluation
of the effective thermal conductivity of quiescent He II is extended to
describe the transition to fully turbulent regime, where the heat flux is
proportional to the cubic root of the temperature gradient (Gorter-Mellink
regime). To do so we use an expression for the quantum vortex line density $L$
in terms of the heat flux considering the influence of the walls. From it, and
taking into account the friction force of normal component against the
vortices, we compute the effective thermal conductivity.
