  A microscopic approach is developed to account for the magnitudes of the
supercurrents observed experimentally in carbon nanotubes placed between
superconducting contacts. We build up a model for the nanotube ropes
encompassing the electron repulsion from the Coulomb interaction and the
effective attraction given by phonon exchange. We show that the available
experimental data are consistent with the expected decay of the supercurrents
along the length of the nanotube samples. Our results stress that the
propagation of the Cooper pairs is favored in the thick ropes, as a consequence
of the reduction in the strength of the Coulomb interaction from the
electrostatic coupling between the metallic nanotubes. We also provide an
explanation for the temperature dependence of the supercurrents observed in the
experiments, remarking the existence of a crossover from a very flat behavior
at low temperatures to a pronounced decay in the long nanotubes.
