We consider an autonomous heat engine in simultaneous contact with a hot and a cold reservoir, and described within a linear-irreversible framework. In the tight-coupling approximation, the rate of entropy generation is effectively written in terms of a single thermal flux which is a homogeneous function of the hot and cold fluxes. The specific algebraic forms of the effective flux are deduced in scenarios containing internal and external irreversibilities for the typical example of a thermoelectric generator.