  In this work, the effect of electron-phonon (e-ph) coupling on both electron
and phonon transport of metals is investigated via first principles
calculations. A Monte-Carlo (MC) approach for solving the coupled
electron-phonon Boltzmann transport equations is developed to investigate
thermal conductivity of metals. In this approach, the anisotropic electron band
structure, phonon dispersion in the full Brillouin zone, and mode-dependent
thermal relaxation time of electrons and phonons are calculated from first
principles. Using this approach, MC simulations of coupled e-ph thermal
transport at different temperatures in {\alpha}-U and Ag are performed. These
two materials were selected as a way to demonstrate the applicability of the
method in different fields. The results indicate that the electron relaxation
time due to phonon scattering is orders of magnitude smaller than the phonon
relaxation time due to electron scattering. The results also show that, in
phonon thermal transport, the impact of ph-e scattering is almost negligible
and the ph-ph scattering dominates phonon transport. At high temperature, the
electrons dominate thermal transport in both {\alpha}-U and Ag. However, at low
temperature, the phonon contribution to the total thermal conductivity of
{\alpha}-U is significant. Moreover, the Lorenz ratio deviates from the
Sommerfeld value at low to intermediate temperatures, where the Wiedemann-Franz
law is not applicable. Finally, we show that the Ag electronic thermal
conductivity shows a stronger size effect than its phonon thermal conductivity.
