Thermodynamic bounds and general properties of optimal efficiency and power in linear responses

We study the optimal exergy efficiency and power for thermodynamic systems with an Onsager-type “current-force” relationship describing the linear response to external influences. We derive, in analytic forms, the maximum efficiency and optimal efficiency for maximum power for a thermodynamic machine described by a $N\times N$ symmetric Onsager matrix with arbitrary integer $N$. The figure of merit is expressed in terms of the largest eigenvalue of the “coupling matrix” which is solely determined by the Onsager matrix. Some simple but general relationships between the power and efficiency at the conditions for (i) maximum efficiency and (ii) optimal efficiency for maximum power are obtained. We show how the second law of thermodynamics bounds the optimal efficiency and the Onsager matrix and relate those bounds together. The maximum power theorem (Jacobi's Law) is generalized to all thermodynamic machines with a symmetric Onsager matrix in the linear-response regime. We also discuss systems with an asymmetric Onsager matrix (such as systems under magnetic field) for a particular situation and we show that the reversible limit of efficiency can be reached at finite output power. Cooperative effects are found to improve the figure of merit significantly in systems with multiply cross-correlated responses. Application to example systems demonstrates that the theory is helpful in guiding the search for high performance materials and structures in energy researches.

Figure 1

The ratio of optimal efficiency at maximum power to the maximum efficiency $\frac{{\varphi }_{\mathrm{opt}}\left({\dot{W}}_{max}\right)}{{\varphi }_{max}}$ (solid curve) and the ratio of the power at maximum efficiency to the maximum power $\frac{\dot{W}\left({\varphi }_{max}\right)}{{\dot{W}}_{max}}$ (dashed curve) as functions of the maximum efficiency ${\varphi }_{max}$, as given by Eq. (21), for thermodynamic machines with a symmetric Onsager matrix.

Figure 2

Schematic of realistic thermodynamic machines. A machine accepts input energy and converts it into output energy. The output can be assigned to a huge reservoir (e.g., an electrical power grid with huge capacity) (a) or to a finite device (b). In realistic situations there are mechanisms that dissipate part of input energy and prevent it from being converted into useful outputs, as well as mechanisms that consume part of output energy and reduce the amount of useful outputs. These mechanisms are called “parasitic dissipation.”

Figure 3

The exergy efficiency $\varphi$ (solid curves), output power $\dot{W}$ (dashed curves), and total entropy production ${\dot{S}}_{\mathrm{tot}}$ (dotted curves) as functions of $x$ for $r=1$ (a), $r=0.6$ (b), and $r=1.2$ (c). For each figure the left region with positive efficiency is the operating region of the machine, while the right region with positive efficiency is the operating region for the reversed machine. The definitions of efficiency and output power are different for the machine and the reversed machine.

Figure 4

Spin-thermoelectric cooling. A spin-thermoelectric (“spin-TE”) material (i.e., a conducting ferromagnetic material) sandwiched between two ferromagnetic electrodes with different temperature $T$, electrochemical potential $\mu \equiv ({\mu }_{\uparrow }+{\mu }_{\downarrow })/2$, and spin accumulation $m\equiv ({\mu }_{\uparrow }-{\mu }_{\downarrow })/\left(2e\right)$, where ${\mu }_{\uparrow }$ and ${\mu }_{\downarrow }$ are the electrochemical potentials for spin-up and spin-down electrons, respectively, and $e$ is the carrier charge. For a setup with $T_h>T_c,{\mu }_h>{\mu }_c$, and $m_h>m_c$ (the subscripts $h$ and $c$ denoting the hot and cold terminals, respectively), cooling (heat flowing from the cold terminal to the hot terminal) is driven by both the charge and the spin flows.

Figure 5

Polar plot of $\xi$ vs $\theta$. The parameters are $P=0.2,P^{\prime }=0.8,S=50$ $\mu \text{V}$/K, and $T=300$ K. The heat conductivity is ${\kappa }_0=\sigma LT$ with the Lorenz number of $L=2.5\times {10}^{-8}$ W $\Omega$ ${\mathrm{K}}^{-2}$. The arrows indicate the relative direction between ${\vec{j}}_{q1}$ (red arrows) and ${\vec{j}}_{q2}$ (green arrows). The red dots represent the thermoelectric figure of merit ${\xi }_{TE}$, the green triangles represent the spin-Peltier figure of merit ${\xi }_{SP}$, and the blue squares denote the figure of merit $\xi$ of combined thermoelectric and spin-Peltier cooling.

Figure 6

(a) The ratio of the figure of merit of spin-Peltier cooling ${\xi }_{SP}$ to that of thermoelectric cooling ${\xi }_{TE}$ as a function of $P^{\prime }$. (b) The enhancement of figure of merit due to cooperative effect, $\xi /\max ({\xi }_{TE},{\xi }_{SP})$, as a function of $P$ and $P^{\prime }$. The parameters are $S=50$ $\mu \text{V}$/K and $T=300$ K. The heat conductivity is ${\kappa }_0=\sigma LT$ with the Lorenz number of $L=2.5\times {10}^{-8}$ W $\Omega$ ${\mathrm{K}}^{-2}$. The white region in (b) near $P=1$ is forbidden by the second law of thermodynamics.

Figure 7

Energy conversion in biological reaction. Biological reactions, $A+B\leftrightarrow C+\text{energy}$ and $Q+\text{energy}+E\leftrightarrow P$, take place in the reaction center. The first reaction produces energy which is stored in material $P$ via the second reaction. At steady states there are continuous flows of materials across the membrane of the reaction center to facilitate continuous reactions. The membrane keeps a density (chemical potential) difference between the reaction center and the outside to control reaction rates. Arrows in the figure indicate possible flows of materials when energy is produced and stored in $P$.