  We model a Brownian heat engine as a Brownian particle that hops in a
periodic ratchet potential where the ratchet potential is coupled with a
linearly decreasing background temperature. It is shown that the efficiency of
such Brownian heat engine is far from Carnot efficiency even at quaistatic
limit. At quasistatic limit, the efficiency of the heat engine approaches the
efficiency of endoreversible engine $\eta=1-\sqrt{{T_{c}/T_{h}}}$ \cite{c18}.
On the other hand, the maximum power efficiency of the engine approaches
$\eta^{MAX}=1-({T_{c}/T_{h}})^{1\over 4}$. Moreover, the dependence of the
current as well as the efficiency on the model parameters is explored
analytically by omitting the heat exchange via the kinetic energy. In this case
we show that the optimized efficiency always lies between the efficiently at
quaistatic limit and the efficiency at maximum power. On the other hand, the
efficiency at maximum power is always less than the optimized efficiency since
the fast motion of the particle comes at the expense of the energy cost. The
role of time on the performance of the motor is also explored via numerical
simulations. Our numerical results depict that the time $t$ as well as the
external load dictate the direction of the particle velocity. Moreover the
performance of the heat engine improves with time. At large $t$ (steady
state),the velocity, the efficiency and the coefficient of performance of the
refrigerator attain their maximum value.
