From maximum power to a trade-off optimization of low-dissipation heat engines: Influence of control parameters and the role of entropy generation

For a low-dissipation heat engine model we present the role of the partial contact times and the total operational time as control parameters to switch from maximum power state to maximum $\mathrm{\Omega }$ trade-off state. The symmetry of the dissipation coefficients may be used in the design of the heat engine to offer, in such switching, a suitable compromise between efficiency gain, power losses, and entropy change. Bounds for entropy production, efficiency, and power output are presented for transitions between both regimes. In the maximum power and maximum $\mathrm{\Omega }$ trade-off cases the relevant space of parameters are analyzed together with the configuration of minimum entropy production. A detailed analysis of the parameter's space shows physically prohibited regions in which there is no longer a heat engine and another region that is physically well behaved but is not suitable for possible optimization criteria.

Figure 1

Comparison between the ${\tilde{P}}_{\text{max}}$ regime (dashed curves, red online) and ${\tilde{\mathrm{\Omega }}}_{\text{max}}$ regime when ${\eta }_{\text{max}}={\eta }_C$ (continous curves, blue online) and when ${\eta }_{\text{max}}\ne {\eta }_C$ (dotted curve, black online; see end of Sec. 3). When ${\tilde{\mathrm{\Sigma }}}_{\mathrm{c}}=\alpha$ then, ${\tilde{t}}_{{\tilde{\mathrm{\Omega }}}_{\text{max}}}$ and ${\tilde{t}}_{{\tilde{P}}_{\text{max}}}$ reach their maximum values [see dots in (a)], meanwhile ${\tilde{P}}_{\text{max}}$ and ${\tilde{\mathrm{\Omega }}}_{\text{max}}$ reach their minimum values [see dots in (c)]. In all cases $\tau =0.2$.

Figure 2

Parametric $\eta -\tilde{P}$ plot at ${\dot{\stackrel{̃}{\mathrm{\Omega }}}}_{\text{max}}$ conditions. Some relevant points are labeled with the corresponding ${\tilde{\mathrm{\Sigma }}}_{\mathrm{c}}, \alpha$, and $\tilde{t}$ values and the associated values of the power and efficiency for $\tau =0.2$.

Figure 3

In (a) the difference of the $\tilde{t}$ values between ${\tilde{P}}_{\text{max}}$ and ${\tilde{\mathrm{\Omega }}}_{\text{max}}$ conditions is depicted; in (b), the corresponding difference of $\alpha$ values. In (c), (d), and (e), the ratio of $\eta , \dot{\stackrel{̃}{\mathrm{\Delta }S}}$, and $\tilde{P}$, respectively, for $\tau =0.2$.

Figure 4

In (a), we show the efficiency $\eta$ with an operation time ${\tilde{t}}_{{\tilde{P}}_{\text{max}}}$. Over this surface the values of $\alpha$ that optimize $\tilde{P}, \tilde{\mathrm{\Omega }}$, and $\dot{\stackrel{̃}{\mathrm{\Delta }S}}$ are shown. The $\alpha -{\tilde{\mathrm{\Sigma }}}_{\mathrm{c}}$ plane of physical relevance is depicted in the shaded area (on the plane $\eta =0$). In (b), the same is done with an operation time ${\tilde{t}}_{{\tilde{\mathrm{\Omega }}}_{\text{max}}}$. As it was shown in Ref. [46] for fixed total time the parametric curves $\eta$ vs. $\tilde{P}$ with a variation of the parameter $\alpha$ produce a loop, which is typical of irreversible processes. In (c), the loops for ${\tilde{\mathrm{\Sigma }}}_{\mathrm{c}}=0.2$ are depicted. Over each loop the location of the $\alpha$ values that optimize $\tilde{P}, \tilde{\mathrm{\Omega }}, \eta$, and $\dot{\stackrel{̃}{\mathrm{\Delta }S}}$ is marked. The dot located at ${\alpha }_{{\eta }_{\text{max}}}$ does not match the ${\alpha }_{{\dot{\stackrel{̃}{\mathrm{\Delta }S}}}_{\text{min}}}$ one unless $\tilde{t}\to \infty$. In (d) and (e) the efficiency surfaces for $\alpha$ constraint to ${\alpha }_{{\tilde{P}}_{\text{max}}}$ and ${\alpha }_{{\tilde{\mathrm{\Omega }}}_{\text{max}}}$, respectively. In both curves as $\tilde{t}\to \infty$ the efficiency tends to ${\eta }_C$. In (f), one can see parabolic curves, typical of endoreversible engines. In all plots $\tau =0.2$.

Figure 5

$\dot{\stackrel{̃}{\mathrm{\Delta }S}}$ for two different values of $\tau$. In (a) we show the case $\tau =1/8$ and in (b) the case $2/5$. In each plot dotted lines depict the cases with ${\tilde{t}}_{{\tilde{P}}_{\text{max}}}$ and continuous lines depict the cases with ${\tilde{t}}_{{\tilde{\mathrm{\Omega }}}_{\text{max}}}$; for each $\tilde{t}$ case from the bottom to the top the curves using the $\alpha$ values that minimize entropy production (green online), maximum efficiency (black online), maximum Omega (blue online), and maximum power output (red online). Depending on the value of $\tau$ both set of curves (dashed and continuous) may have a minimum or not as can be seen in (a) and (b).

Figure 6

In (a) the entropy production for the cases $\tilde{t}=t_{{\tilde{P}}_{\text{max}}}$ (upper sheet, red online) and $\tilde{t}=t_{{\tilde{\mathrm{\Omega }}}_{\text{max}}}$ (lower sheet, blue online) are depicted. In both surfaces the level curves of $\eta =\text{constant}$ are depicted. In (b) a close caption of both surfaces. In both figures $\tau =1/5$. Remarkably, only $\alpha \to {\tilde{\mathrm{\Sigma }}}_{\mathrm{c}}\to 0$ can give the ${\eta }^{+}$'s.