  We present here the first stellar models on the Hertzsprung-Russell diagram
(HRD), in which convection is treated according to the novel scale-free
convection theory (SFC theory) by Pasetto et al. (2014). The aim is to compare
the results of the new theory with those from the classical, calibrated
mixing-length (ML) theory to examine differences and similarities. We integrate
the equations describing the structure of the atmosphere from the stellar
surface down to a few percent of the stellar mass using both ML theory and SFC
theory. The key temperature over pressure gradients, the energy fluxes, and the
extension of the convective zones are compared in both theories. The analysis
is first made for the Sun and then extended to other stars of different mass
and evolutionary stage. The results are adequate: the SFC theory yields
convective zones, temperature gradients of the ambient and of the convective
element, and energy fluxes that are very similar to those derived from the
"calibrated" MT theory for main sequence stars. We conclude that the old scale
dependent ML theory can now be replaced with a self-consistent scale-free
theory able to predict correct results, one which is simpler and more
physically grounded than the ML theory. Fundamentally, the SFC theory offers a
deeper insight of the underlying physics than numerical simulations.
