Entropy generation and unified optimization of Carnot-like and low-dissipation refrigerators

The connection between Carnot-like and low-dissipation refrigerators is proposed by means of their entropy generation and the optimization of two unified, compromise-based figures of merit. Their optimization shows that only a limited set of heat transfer laws in the Carnot-like model are compatible with the results stemming from the low-dissipation approximation, even though there is an agreement of the related physical spaces of variables. A comparison between two operation regimes and relations among entropy generation, efficiency, cooling power. and power input are obtained, with emphasis on the role of dissipation symmetries. The results extend previous findings for heat engines at maximum power conditions.

Figure 1

Sketches of a LD refrigeration (a) and of an irreversible CL RE with a heat leak $Q_L$ (b).

Figure 2

(a) $T_{\mathrm{hw}}$ and $T_{\mathrm{cw}}$ from Eq. (15). Note that as a heat leak appears, both internal temperatures cannot be in thermal equilibrium with the external reservoirs. (b) Total operation time $\tilde{t}({\tilde{Q}}_L,a_{\mathrm{c}})$ according to Eq. (16). In both figures the representative values $\alpha =\frac{1}{5}$, $\tau =0.4$, and ${\tilde{\mathrm{\Sigma }}}_{\mathrm{c}}=\frac{1}{2}$ are considered; the qualitative behavior for other values is the same.

Figure 3

(a) The upper and lower bounds of the COP for the CL RE at maximum-$\chi$ regime vs the heat transfer law exponent $k$. The ${\sigma }_{\mathrm{hc}}\to \{0,\infty \}$ along with the ${\sigma }_{\mathrm{hc}}^{\mathrm{sym}}$ case [Eq. (37)] are plotted. The bounds provided by Eqs. (26) and (27) are labeled. (b) A close view of ${\epsilon }_{{\chi }_{max}}$ using ${\sigma }_{\mathrm{hc}}^{\mathrm{sym}}$ from Eq. (37). In all cases the representative value of $\tau =0.7$ is used.

Figure 4

Influence of the heat leak on the upper and lower bounds of the COP at the ${\chi }_{max}$ regime. The cases ${\sigma }_{\mathrm{L}}=\{0,0.005,0.015\}$ are displayed. The representative value $\tau =0.7$ is used.

Figure 5

COP of the endoreversible ($Q_{\mathrm{L}}=0$) RE operating at ${\mathrm{\Omega }}_{max}$ and two cases with $Q_{\mathrm{L}}=\{0.005,0.015\}$. The representative value of $\tau =0.7$ is used.

Figure 6

Left: ${\tilde{\mathrm{\Omega }}}_{max}$ (dashed lines, blue online) and ${\tilde{\chi }}_{max}$ (continuous lines, red online). Right: Comparison between ${\tilde{\mathrm{\Omega }}}_{max}$ and ${\tilde{\chi }}_{max}$. In all cases $\tau =0.7$.

Figure 7

${\dot{\stackrel{̃}{\mathrm{\Delta }S}}}_{\mathrm{tot}}$ under the time constraints $\tilde{t}={\tilde{t}}_{{\tilde{\chi }}_{max}}$ (upper surface) and $\tilde{t}={\tilde{t}}_{{\tilde{\mathrm{\Omega }}}_{max}}$ (lower surface). Over each surface the level curves denote the configurations of ${\tilde{\mathrm{\Sigma }}}_{\mathrm{c}}$ and $\alpha$ that produce the same COP. We use the representative value $\tau =0.7$.