Effects of dark energy on $P-V$ criticality and efficiency of charged Rotational black hole

Abstract

In this paper, we study $P-V$ criticality of Kerr-Newman $AdS$ black hole with a quintessence field. We calculate critical quantities and show that for the equation state parameter $\omega= -\frac{1}{3}$, the obtained universal ratio ($\frac{P_{c}\upsilon_{c}}{T_{c}}$) is quite same as Kerr-Newman $AdS$ black hole without dark energy parameter. We investigate the influence of quintessence field $\alpha$, equation state parameter $\omega$ and angular momentum $J$ on the efficiency $\eta$. We find that $\eta$ is increased by increasing $J$ and $\alpha$ and decreasing charge $Q$ of black hole. We show when $\omega$ increases from $-1$ to $-\frac{1}{3}$ the efficiency decreases. Also we study ratio $\frac{\eta}{\eta_{C}}$ (which $\eta_{C}$ is the Carnot efficiency) and see that the second law of the thermodynamics is satisfied by special values of $J$ and $\alpha$ and holds for any value of $Q$. We notice that in this case by increasing $\omega$ from $-1$ to $-\frac{1}{3}$ the range of $J$ and $\alpha$ increases.