Closed-loop approach to thermodynamics

We present the closed-loop approach to linear nonequilibrium thermodynamics considering a generic heat engine dissipatively connected to two temperature baths. The system is usually quite generally characterized by two parameters: the output power $P$ and the conversion efficiency $\eta$, to which we add a third one, the working frequency $\omega$. We establish that a detailed understanding of the effects of the dissipative coupling on the energy conversion process requires only knowing two quantities: the system's feedback factor $\beta$ and its open-loop gain $A_0$, which product $A_0\beta$ characterizes the interplay between the efficiency, the output power, and the operating rate of the system. By raising the abstract hermodynamic analysis to a higher level, the feedback loop approach provides a versatile and economical, hence fairly efficient, tool for the study of any conversion engine operation for which a feedback factor can be defined.

Figure 1

Nodal description of a thermodynamic system composed of a conversion engine coupled to two constant temperature baths through heat exchangers.

Figure 2

General description of a linear closed-loop system. The outgoing variable $x_{\mathrm{out}}$ results from a multiplication by $A_0$ of the entering $x_{\mathrm{in}}$, and a feedback contribution of the $\beta$ factor.

Figure 3

Conversion engine as a closed-loop system in (a) the temperature-potential feedback configuration, and (b) the temperature-flux feedback configuration.

Figure 4

Basic electrical description of a thermoelectric system.

Figure 5

Thermoelectric system as a closed-loop system, with (a) temperature-voltage loop, and (b) temperature-current loop.

Figure 6

Voltage (dashed), current (dot dashed), and power (solid) closed-loop gain. Note that the location of the maximum power does not coincide with the traditional assumption $R_{\mathrm{load}}=R_{\mathrm{in}}$ (vertical dashed line on the left side of the panel).

Figure 7

$\omega$ dependence of $arg\left(A_{0X}{\beta }_X\right)/\pi$ with varying ${\omega }_{\kappa }$ in the range $10{}^{-2}$ (smaller-amplitude curve) to $10{}^{-5}$ (larger-amplitude curve), ${\omega }_{\alpha }$ and ${\omega }_{\sigma }$ being fixed to 0.1.

Figure 8

Minimum value of $arg\left(A_{0X}{\beta }_X\right)/\pi$ as a function of ${\omega }_{\kappa }$.