  A thermoelectric generator converts a temperature difference into electrical
energy. Its energy conversion efficiency is determined by the steady-state
temperature distribution inside the generator. By assuming the thermoelectric
material in the generator has a temperature-independent Seebeck coefficient and
the generator is one-dimensional, we show that the second-order
integro-differential equation describing the inside temperature distribution
has a unique solution for any given ratio of external load resistance to the
internal resistance. Hence the efficiency is well defined. Furthermore, we show
the efficiency has an explicit formula in terms of the temperature-dependent
thermal conductivity and electrical resistivity of the thermoelectric material.
On the other hand, if we impose an external load resistance value, not the
ratio, then the integro-differential equation can have multiple solutions.
