Efficiency at maximum power output of an irreversible Carnot-like cycle with internally dissipative friction

We investigate the efficiency at maximum power of an irreversible Carnot engine performing finite-time cycles between two reservoirs at temperatures $T_h$ and $T_c$ $(T_c<T_h)$, taking into account of internally dissipative friction in two "adiabatic" processes. In the frictionless case, the efficiencies at maximum power output are retrieved to be situated between $\eta_{_C}/$ and $\eta_{_C}/(2-\eta_{_C})$, with $\eta_{_C}=1-T_c/{T_h}$ being the Carnot efficiency. The strong limits of the dissipations in the hot and cold isothermal processes lead to the result that the efficiency at maximum power output approaches the values of $\eta_{_C}/$ and $\eta_{_C}/(2-\eta_{_C})$, respectively. When dissipations of two isothermal and two adiabatic processes are symmetric, respectively, the efficiency at maximum power output is founded to be bounded between 0 and the Curzon-Ahlborn (CA) efficiency $1-\sqrt{1-\eta{_C}}$, and the the CA efficiency is achieved in the absence of internally dissipative friction.