  A wide range of unconventional transport phenomena have recently been
observed in single-crystal delafossite metals. Here, we present a theoretical
framework to elucidate electron transport using a combination of
first-principles calculations and numerical modeling of the anisotropic
Boltzmann transport equation. Using PdCoO$_2$ as a model system, we study
different microscopic electron and phonon scattering mechanisms and establish
the mean free path hierarchy of quasiparticles at different temperatures. We
treat the anisotropic Fermi surface explicitly to numerically obtain
experimentally-accessible transport observables, which bridge between the
"diffusive", "ballistic", and "hydrodynamic" transport regime limits. We
illustrate that distinction between the "quasi-ballistic", and
"quasi-hydrodynamic" regimes is challenging and often needs to be quantitative
in nature. From first-principles calculations, we populate the resulting
transport regime plots, and demonstrate how the Fermi surface orientation adds
complexity to the observed transport signatures in micro-scale devices. Our
work provides key insights into microscopic interaction mechanisms on open
hexagonal Fermi surfaces and establishes their connection to the macroscopic
electron transport in finite-size channels.
