Efficiency of nonideal Carnot engines with friction and heat losses

In nonideal thermodynamic engines the efficiency is well below the Carnot efficiency $η=1−T1/T2.$ In 1975 an expression for the efficiency of a nonideal Carnot engine with heat losses was derived, yielding $η=1−T1/T2$ at maximum power output. In this paper, a corresponding relation is obtained for more general nonideal Carnot engines. If there are friction losses only, the result is $η=(1−T1/T2)/2.$ If friction and heat losses are both included, the efficiency at maximum power depends on a dimensionless parameter $λ*$ that takes into account the effects of friction and heat conduction, and can vary between the values obtained for friction and heat losses separately, $(1−T1/T2)/2<ηpmax<1−T1/T2.$ A general relation between efficiency and power output is established, and it is shown that an appreciable gain in efficiency can be obtained at a power output only slightly below its maximum.