  We investigate the energetics of a Brownian motor driven by position
dependent temperature, commonly known as the B{\"u}ttiker-Landauer motor.
Overdamped models (M=0) predict that the motor can attain Carnot efficiency.
However, the overdamped limit ($M\to 0$), contradicts the previous prediction
due to the kinetic energy contribution to the heat transfer. Using molecular
dynamics simulation and numerical solution of the inertial Langevin equation,
we confirm that the motor can never achieve Carnot efficiency and verify that
the heat flow via kinetic energy diverges as $M^{-1/2}$ in the overdamped
limit. The reciprocal process of the motor, namely the B{\"u}ttiker-Landauer
refrigerator is also examined. In this case, the overdamped approach succeeds
in predicting the heat transfer only when there is no temperature gradient. Its
found that the Onsager symmetry between the motor and refrigerator does not
suffer from the singular behavior of the kinetic energy contribution.
