Unregenerated Stirling cycle ($\eta =\frac{nR(T_h-T_c)\ln r}{nR{T}_h\ln r+C_V(T_h-T_c)}$)

For an ideal working gas and $r=V_2/V_1>1$, two isotherms at $T_h,T_c$ produce net work $nR(T_h-T_c)\ln r$. Without a regenerator, isochoric heating requires additional external heat $C_V(T_h-T_c)$, giving the displayed thermal efficiency, strictly below Carnot efficiency. A perfect reversible regenerator recovers this heat and attains the bound instead. Reversing the unregenerated cycle gives cooling coefficient of performance $T_c/(T_h-T_c)-C_V/(nR\ln r)$; positive cooling requires a sufficiently large expansion ratio.