Effects of dark energy on the efficiency of charged AdS black holes as heat engine

In this paper, we study the heat engine where charged AdS black holes surrounded by dark energy is the working substance and the mechanical work is done via $PdV$ term in the first law of black hole thermodynamics in the extended phase space. We first investigate the effects of a kind of dark energy (quintessence field in this paper) on the efficiency of the RN-AdS black holes as heat engine defined as a rectangle closed path in the $P-V$ plane. We get the exact efficiency formula and find that quintessence field can improve the heat engine efficiency which will increase as the field density $\rho_q$ grows. At some fixed parameters, we find that bigger volume difference between the smaller black holes($V_1$) and the bigger black holes($V_2$ ) will lead to a lower efficiency, while the bigger pressure difference $P_1-P_4$ will make the efficiency higher but it is always smaller than 1 and will never be beyond Carnot efficiency which is the maximum value of the efficiency constrained by thermodynamics laws, this is consistent to the heat engine in traditional thermodynamics. After making some special choices for thermodynamical quantities, we find that the increase of electric charge $Q$ and normalization factor $a$ can also promote heat engine efficiency which would infinitely approach the Carnot limit when $Q$ or $a$ goes to infinity.