In the past few decades, neutrinos, which are predicted to be massless particles by the Standard Model of Particle Physics, have been shown to have non-zero mass. The absolute scale of this neutrino mass has significant implications in particle physics, astrophysics, and cosmology. The KATRIN experiment is designed to measure this absolute scale by examining the beta decay spectrum of molecular, gaseous tritium source. In this thesis, the beta decay of this molecular tritium is simulated to study the effects of "Doppler shifts" in the energy of the emitted electrons due to the random thermal motion and fluid flow velocity of the differentially pumped tritium gas. Simulated spectra are presented for three different neutrino masses and the relative effects of the thermal and flow velocities are discussed.